Subspaces and Quotients of Topological and Ordered Vector Spaces

Home , Bornivorous set, Quasibarrelled space, Saturated family, Topological homomorphism

Zoran Kadelburg Stojan Radenovi´c

SUBSPACES AND QUOTIENTS OF TOPOLOGICAL AND ORDERED VECTOR SPACES

Novi Sad, 1997. CONTENTS

INTRODUCTION...... 1

I: TOPOLOGICAL VECTOR SPACES...... 3 1.1. Some properties of subsets of vector spaces ...... 3 1.2. Topological vector spaces...... 6 1.3. Locally convex spaces ...... 12 1.4. Inductive and projective topologies ...... 15 1.5. Topologies of uniform convergence. The Banach-Steinhaus theorem 21 1.6. Duality theory ...... 28

II: SUBSPACES AND QUOTIENTS OF TOPOLOGICAL VECTOR SPACES ...... 39 2.1. Subspaces of lcs’s belonging to the basic classes ...... 39 2.2. Subspaces of lcs’s from some other classes ...... 47 2.3. Subspaces of topological vector spaces ...... 56 2.4. Three-space-problem for topological vector spaces...... 60 2.5. Three-space-problem in Fr´echet spaces...... 65

III: ORDERED TOPOLOGICAL VECTOR SPACES ...... 72 3.1. Basics of the theory of Riesz spaces...... 72 3.2. Topological vector Riesz spaces ...... 79 3.3. The basic classes of locally convex Riesz spaces ...... 82 3.4. l-ideals of topological vector Riesz spaces ...... 90 3.5. 3SP in the class of locally convex Riesz spaces ...... 92 IV: SUPPLEMENT: ENLARGEMENTS OF LOCALLY CONVEX TOPOLOGIES ...... 100 4.1. Finite-dimensional enlargements ...... 100 4.2. Countable enlargements of barrelled topologies ...... 103 4.3. Satisfactory subspaces of barrelled spaces ...... 106 4.4. Open problems...... 112

REFERENCES...... 113

SUBJECT INDEX ...... 120 INTRODUCTION

First results concerning not necessarily normed topological vector spaces date back to middle thirties of this century. But it was not until late forties and early fifties that the theory of such spaces was formed as a special branch in functional analysis. This was the time when it became clear that a lot of applications, in particular the theory of distributions, need more abstract approach than the classical Banach one. Within the framework of the new categories of topological vector, in particular locally convex, spaces and, a bit later, of ordered topological vector spaces, a lot of special classes of spaces were introduced. The reason was mainly that the three fundamental principles of linear analysis, in particular the uniform boundedness principle and the closed graph and open mapping theorems, needed special types of spaces to serve as domains and/or codomains for respective mappings. Such classes of spaces, e.g. barrelled, quasibarrelled, bornological and many others, proved themselves to be important in connection with other problems, too, such as problems in duality theory, reflexivity, tensor products and so on. One of the important questions concerning these classes of spaces was their stability in connection with subspaces and quotients. While most of them (but not all) are stable with respect to quotients by closed subspaces, few are stable with respect to arbitrary, even closed subspaces. So it was natural to search for various sufficient conditions which would imply stabilities of such kind. First results of that type were obtained in early fifties, but investigations have continued and there are a lot of interesting recent results. Some questions even in pure Banach setting are still open, e.g. “is there a Banach space of infinite dimension such that all of its dense subspaces are barrelled” or “does every infinite-dimensional Banach space have a separable quotient”? This book is divided into four chapters. The first one contains a very brief introduction to the theory of topological vector spaces. The intention was to give only the fundamental ideas of the theory, particularly those which will be used in the rest of the book. Some important theorems are stated without proofs, and the reader is referred to the classical books covering these problems. But we hope that this will be enough to get prepared for reading the remaining three chapters. Especially, at least formally speaking, the prerequisites for the whole book are just the basics of linear algebra and general topology, although, of course, the acquaintance with classical functional analysis is desirable. 2 Introduction

Chapter II treats thoroughly the problems of subspaces and quotients for various classes of topological vector, in particular locally convex, spaces. Also, a kind of a converse question, the so-called three-space-problem, is investigated. Some of the results of the authors are included in this and the following chapter. In many applications of functional analysis, besides linear and topological structures, also the structure of order is often important. So a special theory of ordered vector spaces, and in particular ordered topological vector spaces, has been developed. Our third chapter treats the basics of this theory and in particular those concepts which are close to the ones described in the ﬁrst part of the book. It is shown that the richer structure enables more positive results than in the category of vector spaces without order. There are a lot of problems and streams of investigation which are actual nowadays in the theory of topological vector spaces. We chose some of them and included as a supplement in the chapter IV. We hope that this will help to get an idea in what directions the theory may be developed and, maybe, some of the readers will try to do something on their own in this ﬁeld. A brief list of open questions which is given at the end might help. Although the list of references is rather long, it is very far from being complete, even when the special topics of this book are concerned. We mentioned just those titles which we referred to directly and, with a few exceptions, we didn’t mention articles which are already included in existing monographs. We are very grateful to the Institute of Mathematics of the University of Novi Sad for publishing this monograph.

Beograd, October 1996 Authors I. TOPOLOGICAL VECTOR SPACES

1.1. SOME PROPERTIES OF SUBSETS OF VECTOR SPACES

All vector space in this book are supposed to be defined over the fixed field K, which is either the field R of real or the field C of complex numbers. Only in the third chapter just real scalars are used. In this paragraph we shall describe some properties of subsets of vector spaces which are important in the theory of topological and ordered vector spaces. Let E be a vector space (over the field K) and let A be its subset. We say that A is: 1◦ circled if λA ⊂ A for each λ ∈ K for which |λ| 6 1; 2◦ convex if λx + (1 − λ)y ∈ A for each x, y ∈ A and each λ ∈ K, 0 6 λ 6 1; 3◦ absolutely convex if λx + µy ∈ A for each x, y ∈ A and each λ, µ ∈ K for which |λ| + |µ| 6 1. It is clear that the intersection of an arbitrary family of (absolutely) convex subsets is again an (absolutely) convex set. The (absolutely) convex cover of the set A ⊂ E is the intersection of all (absolutely) convex sets containig A; we shall denote it by co A (resp. ΓA). If (Ai), Si ∈ I is a familyS of subsets of E , we shall write coi Ai (resp. ΓiAi ) instead of co( i Ai) (resp. Γ( i Ai)). Pn Proposition 1.1. co A is the set of all elements of the form λixi for 1 Pn Pn λi > 0, λi = 1, xi ∈ A. ΓA is the set of all elements of the form λixi for 1 1 Pn λi ∈ K, |λi| 6 1, xi ∈ A. 1 Proof. We shall prove the assertion for the convex cover; for the absolutely convex cover the proof is similar. It can be checked directly that the elements of the given form create a set which is convex and contains A. Let us show that, conversely, these elements belong to each convex set which contains A. nP−1 Suppose that the last assertion is proved for each sum of the form λixi . 1 Pn nP−1 Let µixi be a combination of the given form and let µi = µ > 0 (for µ = 0 1 1 4 I. Topological vector spaces

n−1 P µi the assertion is trivial). By the inductive hypothesis, y = xi ∈ co A, and so 1 µ Pn µy + (1 − µ)xn = µixi ∈ co A since 0 < µ 6 1. 1 By the circled cover of a set A ⊂ E we understand the intersection of all circled subsets of E which contain A.

Proposition 1.2. The absolutely convex cover of a subset is the convex cover of its circled cover. A set is absolutely convex if and only if it is convex and circled. The circled cover of the convex cover of a set need not be convex. Proof. Since, obviously, each absolutely convex set is circled, it is enough to prove that the convex cover of a circled set A is absolutely convex. According to Proposition 1.1, we have co A ⊂ ΓA. Conversely, let λi 6= 0 from K be given, such Pn Pn Pn that |λi| 6 1 and let xi ∈ A. Denote µi = (λi/|λi|) |λi| and νi = |λi|/ |λi|. 1 1 1 Pn Pn Then each x = |λi|xi ∈ ΓA can be written in the form x = νiµixi , where 1 1 Pn νi = 1, νi > 0 and |µi| 6 1, µixi ∈ A, so that x ∈ co A. Thus ΓA ⊂ co A. 1 Let A = {(0, 0), (1, 0), (1, 1)}, treated as a subset either of R2 or C2 . It is easy to check that the circled cover of the set co A is not convex.

◦ Proposition 1.3. 1 If A1 , ... , An are (absolutely) convex sets and λ1 , ... , λn ∈ K, then λ1A1 + ··· + λnAn is an (absolutely) convex set. 2◦ If E and F are vector spaces over K and L: E → F is a linear mapping, then the image (resp. inverse image) of each (absolutely) convex subset of E (resp. F ) is an (absolutely) convex subset of F (resp. E ). Proof. The assertion 1◦ can be checked directly. Let A be a convex subset of E . Since L(λx + (1 − λ)y) = λLx + (1 − λ)Ly , the set L(A) is convex in F if A is a convex subset of E . Similarly, if B is convex in L(E) and if Lx, Ly ∈ B , then L(λx + (1 − λ)y) ∈ B and so L−1(B) is a convex set in E . For absolutely convex subsets the proof is similar. Let A and B be subsets of a vector space E . We say that A absorbs B if there exists α > 0, such that B ⊂ λA for each λ ∈ K such that |λ| > α. The set A is absorbing in E if it absorbs all the points, more precisely all the singletons from E , i.e. if for each x ∈ E there is α > 0 so that x ∈ λA for each λ ∈ K such that |λ| > α. It can be immediately checked that the following is valid.

Proposition 1.4. Let A and B be circled sets. Then A absorbs B if and only if there exists λ > 0 such that λB ⊂ A. In particular, the set A is absorbing if and only if for each x ∈ E there exists λ > 0 such that λx ∈ A.

We call a sequence U = (Un) of subsets of the vector space E a string if all the subsets Un are circled and absorbing and Un+1 + Un+1 ⊂ Un holds for each 1.1. Some properties of subsets of vector spaces 5

T n ∈ N. The set Un is the n-th knot of this string and N(U) = n∈N Un is its kernel — it is obviously a vector subspace of the space E . To each absorbing absolutely convex set U one can naturally correspond a n−1 string U = (Un) by Un = (1/2 )U , n ∈ N. However, in general, knots of a string need not be convex.

Example. Let E be a vector space and {eα}α∈A its algebraic (Hamel) basis. p p For a positive real number p let us deﬁne a string U = (Un) by

p P P p n−1 Un = { x | x = λαeα, |λα| 6 1/2 }, for 0 < p 6 1, α∈A α∈A µ ¶1/p p P P p n−1 Un = { x | x = λαeα, |λα| 6 1/2 }, for p > 1, α∈A α∈A where in all given sums we assume that λα 6= 0 only for ﬁnitely many indices α. p p For each p we have N(U ) = {0}. For p > 1 the knots Un are obviously convex, but for 0 < p < 1 it need not be true. More precisely, the following is valid. Proposition 1.5. Let 0 < p < 1. Then the following conditions are equivalent: (i) E has an uncountable algebraic dimension; p p (ii) no knot Un of the string U contains an absorbing absolutely convex set.

Proof. (i) =⇒ (ii). Let (eα)α∈A be a basis of E . It is enough to prove (ii) for p n = 1. If there exists an absolutely convex and absorbing set U ⊂ U1 , then there exist εα > 0 such that εαeα ∈ U for all α ∈ A. As A is uncountable, we can ﬁnd ε > 0 such that the set Aε := { α ∈ A | εα > ε } is inﬁnite. If α1, . . . , αn ∈ Aε , n ε P ε p we have εαi eαi ∈ U and eαi ∈ U . On the other hand, U ⊂ U1 and so εαi i=1 n Pn ³ ε ´p 1 > = n1−pεp . Since n is arbitrary, this is a contradiction. i=1 n (ii) =⇒ (i). Let us show that in the countable case each string U = (Un) contains an absolutely convex string. Let {en}n∈N be an algebraic basis of the (1) (1) (n) (n) 1 (n−1) space E . If we choose εi > 0 with εi ei ∈ U1+i , εi with 0 < εi 6 2 εi ∞ (n) P (n) and εi ei ∈ Un+i for n > 1, then the sets Vn := εi [ei]1 form an absolutely i=1 convex string V for which V ⊂ U (i.e. Vn ⊂ Un for each n). Namely,

∞ ∞ P (n) P Vn = εi [ei]1 ⊂ Un+i ⊂ Un. i=1 i=1 P Here we used the following notation: if Mα ⊂ E , α ∈ A, then Mα is the set µ ¶ α∈A S P e⊂A Mα , where e runs through all ﬁnite subsets of the index set A. If α∈e x ∈ E and λ > 0, [x]λ := { µx | |µ| 6 λ }. 6 I. Topological vector spaces

We shall assume that the concepts such as subspace, quotient,(algebraic) dimension and codimension are known to the reader. Particularly, we shall call subspaces of a vector space E which have the codimension 1 hyperplanes (sometimes we shall use this name also for linear manifolds which are obtained by their translation for a ﬁxed vector). If E is a vector space over K, then the set of all linear mappings f : E → K, with linear operations canonically introduced, is called the algebraic dual space of the space E and is denoted by E∗ . An obvious connection between the algebraic dual and hyperplanes is given by Proposition 1.6. To each element f , not identically equal to zero, of the algebraic dual E∗ of a vector space E , a hyperplane H = f −1(0) of this space is corresponded. Conversely, to each hyperplane H of a space E one can correspond an element f ∈ E∗ and α ∈ K such that H = f −1(α).

1.2. TOPOLOGICAL VECTOR SPACES

Definition 1. Let E be a vector space over the field K. Topology t on E is called vector (linear) if it is compatible with linear operations, i.e. if (x, y) 7→ x + y and (λ, x) 7→ λx are continuous mappings on E × E and K × E , respectively. The space E equipped with a Hausdorff linear topology t on it, i.e. the ordered couple (E, t) is called a topological vector space (tvs) or linear topological space. An immediate consequence of the definition is the following

Proposition 2.1. Let (E, t) be a tvs, x0 ∈ E and α ∈ K. Then: ◦ 1 the mapping x 7→ x+x0 is a topological homeomorphism of the space (E, t) onto itself; ◦ 2 if U is a base of neighbourhoods of the point 0 ∈ E , then the sets x0 + U , U ∈ U form a base of neighbourhoods of the point x0 ; 3◦ a linear mapping f : E → F between tvs’s is continuous on E if and only if it is continuous at the point 0. Proposition 2.2. Each neighbourhood of the origin in a tvs (E, t) is an absorbing set. The space (E, t) possesses a base of neighbourhoods of the origin formed of circled subsets and to each circled neighbourhood of the origin U there corresponds a string U = (Un), such that U1 = U and all Un ’s are neighbourhoods of the origin (such a string is called topological). The intersection of all the neighbourhoods of the origin is the set {0}. Proof. If a neighbourhood U of the origin in (E, t) were not absorbent, a point 1 x0 ∈ E would exist which does not belong to any nU , n ∈ N. Then n x0 ∈/ U for 1 all n ∈ N would be valid, which is contrary to the condition n x0 → 0, which is a direct consequence of the continuiuty of the multiplication by scalars. 1.2. Topological vector spaces 7

The continuity of the function λx at the point (0, 0) also implies that for each neighbourhood of the origin U there exist δ > 0 and a neighbourhoodS of the origin W such that λx ∈ U for all x ∈ W and all λ, |λ| 6 δ . Thus, V = { λW | |λ| 6 δ } is a neighbourhood of the origin which is obviously circled and contained in U .

The existence of a string (Un) with the described properties is a consequence of the continuity of the map (x, y) 7→ x + y . The last assertion of the proposition follows because the topology t is Hausdorﬀ. Let us show now that in a certain sense the converse of the previous proposition holds.

Theorem 2.3. Let F be a family of strings on a vector space E such that: for each two strings U, V ∈ F there exists W ∈ F with W ⊂ U ∩ V (i.e. Wn ⊂ Un ∩ Vn for each n ∈ N). Then the knots of all the strings from F form a base of neighbourhoods of the origin of a linear topology t on E (this topology will be denoted as tF and we shall say that it is generatedTby the family F ). The topology tF is Hausdorﬀ (i.e. (E, t) is a tvs) if and only if U∈F N(U) = {0}. Proof. Let U(0) be the family created of all the knots of the strings from F . If the family U(x0) for a point x0 ∈ E is deﬁned as U(x0) = { x0 + U | U ∈ U(0) }, it is clear that all sets from the families U(x0), x0 ∈ E will form a base of neighbourhoods of a certain topology on E . Let us prove that this topology is linear.

The continuity of the mapping (x, y) 7→ x + y at the point (x0, y0) follows from (x0 + V ) + (y0 + V ) ⊂ x0 + y0 + U , where V ∈ U(0) is chosen such that V + V ⊂ U ∈ U(0).

Suppose that a neighbourhood λ0x0 +U ∈ U(λ0x0) is given and let |λ0| 6 n ∈ N. We can ﬁnd V ∈ U(0) such that V + V + ··· + V ⊂ U ; then also nV +V +V ⊂ | {z } n+2 U . Choose an integer m such that x0 ∈ mV . If now |λ0−λ| < 1/m and x ∈ x0+V , then the relation λx ∈ λ0x0 + V + V + nV ⊂ λ0x0 + U follows from

λx = λ0x0 + (λ − λ0)x0 + (λ − λ0)(x − x0) + λ0(x − x0)

1 1 since m (mV ) = V ,(λ − λ0)(x − x0) ∈ m V ⊂ V and λ0(x − x0) ∈ nV . In such a way we have proved that the mapping λx is continuous at the point (λ0, x0). The last assertion can be checked immediately. Examples. 1◦ The simplest examples of tvs’s are normed, in particular Banach spaces. In such a space, a base of neighbourhoods of the origin is formed by the balls Kn = { x ∈ E | kxk < 1/n }. 2◦ The family of all strings in a vector space E generates the topology on E which is the ﬁnest linear topology on this space. It will be denoted as tb(E) . 8 I. Topological vector spaces

◦ 3 We call a tvs (E, tm(E)) a tvs of minimal type if there is no Hausdorff linear topology t on E such that t < tm(E) . 4◦ Let X be a nonempty topological space. The set of all continuous functions X f : X → K, such that supx∈K |f(x)| is finite, creates a subset CK(X) of the set K which is a vector space related to the “coordinate” addition and multiplication by scalars. The sets Un = { f | supx∈X |f(x)| < 1/n } form a base of neighbourhoods for a topology in which CK(X) is a tvs. The examples 1 and 4 are tvs’s in which the topology is metrizable. In the sequel we shall describe such spaces more precisely. Definition 2. Let E be a vector space. A function f : E → R+ is called an F -seminorm on E if: (i) f(x + y) 6 f(x) + f(y) for x, y ∈ E ; (ii) f(λx) 6 f(x) for λ ∈ K, |λ| 6 1, x ∈ E ; (iii) f(λx) → 0 when λ → 0, for fixed x ∈ E . If, moreover, (iv) f(x) = 0 ⇐⇒ x = 0, the F -seminorm f is called an F -norm. Theorem 2.4. For a tvs (E, t) the following conditions are equivalent: 1◦ topology t is metrizable, i.e. there exists a metric d on E which is translation-invariant (d(x, y) = d(x+z, y +z)) and which generates the topology t; 2◦ topology t is Hausdorff and has a countable base of neighbourhoods of the origin; ◦ 3 there exists a string U on E such that N(U) = {0} and t = tU (i.e. t = tF for F = {U}); 4◦ the space (E, t) is F -normed, i.e. there exists an F -norm f on E which generates the topology t. Proof. 1◦ =⇒ 2◦ . If d is a metric generating the topology t on E , a base of neighbourhoods of the origin is formed, for example, by the sets K1/n = { x ∈ E | d(x, 0) < 1/n }, n ∈ N. ◦ ◦ 2 =⇒ 3 . As a string which generates the topology t we can take U = (Un), where Un ⊂ K1/n(0). ◦ ◦ 3 =⇒ 4 . Let U = (Un) be a string with the described properties. In order to construct an F -norm f , consider a dyadic rational δ > 0 of the form ½ ∞ P εk 1, for at most finitely many k ∈ N, δ = n + k , n ∈ N ∪ {0}, εk = k=1 2 0, otherwise.

For such a number δ deﬁne the set Pn P∞ Wδ := U1 + εkUk+1, 1 k=1 1.2. Topological vector spaces 9

and for x ∈ E let us put f(x) := inf{ δ | x ∈ Wδ }. Let us prove that f is an F -norm on the space E .

(i) For numbers δ1 , δ2 we have Wδ1 + Wδ2 ⊂ Wδ1+δ2 and hence x ∈ Wδ1 and y ∈ Wδ2 imply f(x + y) 6 δ1 + δ2 , wherefrom (i) follows. (ii) follows since all the sets Wδ are circled.

(iii) Since all the sets Un are circled and absorbing, if λn → 0, for a given k ∈ N there exists n0 ∈ N such that λnx ∈ Uk+1 for n > n0 . This means that k f(λnx) 6 1/2 for n > n0 .

(iv) Let x ∈ E and f(x) = 0. Then x ∈ Wδ for each δ > 0, particularly x ∈ Uk for each k ∈ N. Hence x ∈ N(U) = {0}. That the constructed F -norm f generates the topology t, follows from the relation

(∗) K1/2n (0) ⊂ Un+1 ⊂ K1/2n (0),

n where we have put K1/2n = { x ∈ E | f(x) < 1/2 } and K1/2n = { x ∈ E | f(x) 6 n 1/2 }. For a proof of the relation (∗), consider an arbitrary x ∈ K1/2n (0), i.e. ∞ n P εk |f(x)| < 1/2 . Then there exists δ = k with εk = 0 for 1 6 k 6 n, such k=1 2 ∞ P n that x ∈ Wδ = εkUk+1 ⊂ Un+1 . For x ∈ Un+1 we have f(x) 6 1/2 , and k=n+1 so x ∈ K1/2n (0). 4◦ =⇒ 1◦ . If f is an F -norm on E , we can define a metric d(x, y) := f(x − y). This metric is obviously translation-invariant and generates the topology t. A metrizable and complete tvs is called a Fréchetspace (an (F )-space). Example. Let Lp be the set of all measurable functions x(t) on the interval R b p I = [a, b] ⊂ R, such that a |x(t)| dt < ∞, where, to the contrary of the standard Banach case, 0 < p < 1 (as usual, we identify functions which are equal a.e.). Define a function f on Lp by

Z b f(x) = |x(t)|p dt. a It can be easily checked that it is an F -norm, and so Lp , equipped with the topology generated by this F -norm, is a metrizable tvs. Remark that the function f is not a norm, and so in this case Lp is not a normed space. In the next proposition we state some properties of open, closed and compact subsets of tvs’s. Since they are very simple, we shall prove only some of them. Proposition 2.5. Let (E, t) be a tvs. Then: 1◦ Each neighbourhood of the origin in E contains a closed neighbourhood, i.e. there is a base of neighbourhoods of the origin formed by closed sets. 10 I. Topological vector spaces

◦ 2 For every set A ⊂ E andT every base U of neighbourhoods of the origin in E , the closure A is given by A = { A + U | U ∈ U }. 3◦ If A is an open and B an arbitrary subset of E , then the set A + B is open. 4◦ If A is a compact and B a closed subset of E , then the set A+B is closed. 5◦ If A is a circled (convex, absolutely convex) subset of E , then its closure A is circled (convex, absolutely convex). Proof. 1◦ For each neighbourhood of the origin U there is a neighbourhood of the origin V such that V + V ⊂ U . Since y ∈ V if and only if (y − V ) ∩ V is nonempty, it follows that V ⊂ V + V ⊂ U . ◦ 4 We shall show that for each x0 ∈/ A + B there exists such a neighbourhood of the origin U that (x0 −U)∩(A+B) = ∅, or, equivalently, (B+U)∩(x0 −A) = ∅. If it were not true, then the intersections (B +U)∩(x0 −A) would form a base of a ﬁlter on x0 −A, when U runs through a base of neighbourhoods of the origin in E . Since A is compact, there would exist a cluster point of this ﬁlter z0 ∈ x0 − A, which would belong also to B + U , and so to B + U + U for each U . As the sets of the form U + U form a base of neighbourhoods of the origin, from 2◦ it would follow that z0 ∈ B , which would bring us to a contradiction. 5◦ We shall prove this assertion for absolutely convex sets. Let C ⊂ E be such a set, let x0, y0 ∈ C and λ, µ ∈ K, |λ| + |µ| 6 1. For a given neighbourhood of the origin U , let V be a circled neighbourhood of the origin for which V +V ⊂ U . For the given points x0 , y0 choose points x, y ∈ C such that x0 − x ∈ V , y0 − y ∈ V . Then

(λx0 + µy0) − (λx + µy) = λ(x0 − x) + µ(y0 − y) ∈ V + V ⊂ U, which means that λx0 + µy0 ∈ C and so C is an absolutely convex set. If A is a subset of a tvs (E, t), then the intersection of all closed (absolutely) convex subsets of E which contain A is called the closed (absolutely) convex cover of the set A. It is equal to the closure of the (absolutely) convex cover of A. The topological dual space of a tvs (E, t), i.e. the set of all continuous linear functionals on E , will be denoted by E0 . The next proposition is a reﬁnement of the Proposition 1.6.

Proposition 2.6. Each hyperplane H in a tvs E is either closed or dense in E ; besides, H = f −1(0) is closed if and only if f is continuous, i.e. f ∈ E0 . Proof. If a hyperplane H is not closed, it has to be dense in E , because if it were not the case, its closure H would be a proper subspace of E which contains (as a proper subset) a maximal subspace H , which is impossible. It is clear that in the case of continuity of f , the subspace H = f −1(0) is closed. Conversely, if H is closed, then E/H is a (Hausdorﬀ) tvs of the dimension 1, and the mapping f can be written in the form f = g◦q , where q : E → E/H is the quotient mapping and g : E/H → K is an isomorphism, and so f is continuous. 1.2. Topological vector spaces 11

Definition 3. 1◦ A subset B of a tvs (E, t) is bounded if it is abosrbed by all neighbourhoods of the origin in (E, t). 2◦ A subset P of a tvs (E, t) is precompact if for each neighbourhood of the origin U in (E, t) there exists a ﬁnite set P0 ⊂ P such that P ⊂ P0 + U . The family of all bounded (precompact) subsets of a tvs (E, t) will be denoted by B(E) (resp. P(E)). If M is an arbitrary family of bounded subsets in (E, t) (M ⊂ B ), we shall say that its subfamily N is fundamental for M if for each set M ∈ M there exists a set N ∈ N such that M ⊂ N . We shall prove some of the following simple properties of bounded and precompact sets.

Proposition 2.7. 1◦ Each subset of a bounded (precompact) set is again bounded (precompact). 2◦ If A is bounded (precompact), then so are A and λA for each λ ∈ K. 3◦ If A and B are bounded (precompact), then so are A ∪ B and A + B . ◦ 4 A subset B of a tvs (E, t) is bounded if and only if for each sequence (xn) in B and an arbitrary sequence of scalars αn → 0, the sequence αnxn tends to the origin in (E, t). 5◦ Each precompact set is bounded. The circled cover of a bounded set is bounded. 6◦ A continuous linear image of a bounded (precompact) set is bounded (precompact). Proof. 4◦ Let B be a bounded set and U a circled neighbourhood of the origin. There exists λ > 0 such that for each n we have xn ∈ λU . Then αnxn ∈ αnλU = |αn|λU . If n0 is big enough, we shall have |αn|λ 6 1 for n > n0 and so αnxn ∈ U for n > n0 . Since this is true for each neighbourhood of the origin U , we have αnxn → 0. Conversely, if a set B is unbounded, for some neighbourhood of the origin U 1 there is a sequence of points xn ∈ B such that xn ∈/ nU . The sequence n xn then obviously does not tend to the origin. 5◦ Let P be a precompact set, U an arbitrary neighbourhood of the origin, V such a neighbourhood of the origin that V + V ⊂ U and P0 a ﬁnite subset of P such that P ⊂ P0 + V . If we choose λ > 1 such that P0 ⊂ λV , we shall have P ⊂ λV + V ⊂ λU . For a tvs which possesses a bounded neighbourhood of the origin we say that it is locally bounded. Such a space possesses a base of neighbourhoods of the origin formed of bounded sets. The spaces Lp , 0 < p < ∞ are examples of such spaces.

Proposition 2.8. Each locally bounded tvs is metrizable.

Proof. Let V be a bounded neighbourhood of the origin in (E, t) and (λn) a sequence of scalars, distinct from 0, such that λn → 0. If U is an arbitrary circled neighbourhood of the origin, then there exists λ ∈ K such that V ⊂ λU since V 12 I. Topological vector spaces

is bounded. If we choose n ∈ N such that |λnλ| 6 1, then λnV ⊂ U , because U is circled. From here it follows that { λnV | n ∈ N } is a base of neighbouhoods of the origin for the topology t, hence this topology is metrizable. Without the proof we state the following Theorem 2.9. Each locally precompact tvs (i.e. a tvs having a precompact neighbourhood of the origin) is of ﬁnite dimension. On the basis of the Proposition 2.5.1◦ we conclude that each tvs is a regular topological space. Moreover, it can be easily proved that each tvs E is a uniform space whose base of vicinities is created by the sets of the form N = { (x, y) | x − y ∈ V }, where V runs through a base U of neighbourhoods of the origin in E . This uniformity is translation-invariant in the sense that

(x, y) ∈ N ⇐⇒ (x + z, y + z) ∈ N for each N of the given form and for each z ∈ E . Taking this into account, we conclude that in topological vector spaces we can speak about such concepts as Cauchy filters (sequences), completeness, completion, uniform continuity, etc. For example, a filter F in E is a Cauchy filter if for each neighbourhood of the origin U in E there is F ∈ F such that F − F ⊂ U . Moreover, as a supplement to the proposition 2.7 we have the following properties which are valid in each uniform space. Proposition 2.10. 1◦ Each Cauchy sequence in a tvs is precompact. 2◦ A subset of a (Hausdorff) tvs E is precompact if and only if its closure in the completion E˜ is compact. 3◦ A subset of a tvs is compact if and only if it is precompact and complete.

1.3. LOCALLY CONVEX SPACES

The most important type of topological vector spaces are locally convex spaces. Definition 1. A topological vector space (E, t) is called a locally convex space (lcs) if it has a base of neighbourhoods of the origin consisting of convex sets. It is clear that each lcs has a base of neighbourhoods of the origin which consists of absolutely convex and closed sets. In the Theorem 2.3 we saw that the topology of a tvs can be described by a family of F -seminorms. For locally convex spaces the corresponding proposition can be stated using seminorms. Definition 2. (a) A nonnegative real function p on a vector space E is called a seminorm if: 1.3. Locally convex spaces 13

1◦ p(x + y) 6 p(x) + p(y) for x, y ∈ E ; 2◦ p(λx) = |λ|p(x) for x ∈ E , λ ∈ K. (b) Let M be an absolutely convex and absorbing subset of a vector space E . The function x 7→ pM (x) = inf{ λ > 0 | x ∈ λM } is called the Minkowski functional of the set M . Proposition 3.1 Let E be a vector space. A real function p on E is a seminorm if and only if p is the Minkowski functional of a certain absolutely convex and absorbing set M ⊂ E . Proof. Let p be a seminorm on E and let M = { x | p(x) < 1 }. It can be easily checked that from the properties 1◦ and 2◦ it follows that the set M is ◦ absolutely convex and absorbing. Let us prove that p = pM . It follows from 2 that { x | p(x) < λ } = λM for each λ > 0. Hence, if p(x) = α, then x ∈ λM for all λ > α, but not for λ < α, and so we obtain

p(x) = inf{ λ > 0 | x ∈ λM } = pM (x).

Conversely, let M be an absolutely convex and absorbing set and let p = pM . If x, y ∈ E and λ1 > p(x), λ2 > p(y), then x + y ∈ λ1M + λ2M . Since M is convex, we have · ¸ λ1 λ2 λ1M + λ2M = (λ1 + λ2) M + M ⊂ (λ1 + λ2)M, λ1 + λ2 λ1 + λ2 wherefrom it follows p(x+y) 6 λ1 +λ2 , and so p(x+y) 6 p(x)+p(y). For proving the property 2◦ , remark that, on the base of circledness of M , the condition λx ∈ µM is equivalent to |λ|x ∈ µM . Thus, if λ 6= 0, then

1 p(λx) = inf{ µ > 0 | x ∈ µM } = inf { |λ|µ | x ∈ µM } = |λ|p(x), |λ| µ>0 which proves 2◦ since p(0) = 0.

Observe that, for the given seminorm p, a set M for which p = pM is not uniquely determined. In fact, the following is valid. Let M be an absolutely convex and absorbing set in a vector space E and let p be a seminorm on E . Then p = pM if and only if { x | p(x) < 1 } ⊂ M ⊂ { x | p(x) 6 1 }. Furthermore, observe that for the given seminorm p on a tvs E , the following conditions are equivalent: 1◦ p is continuous at the origin; ◦ 2 M0 = { x | p(x) < 1 } is an open set in E ; 3◦ p is uniformly continuous on E . The following analogon of the Theorem 2.3 can be deduced easily using these observations. 14 I. Topological vector spaces

Theorem 3.2. Let { pi | i ∈ I } be an arbitrary family of seminorms on a vector space E with the property that for each x0 6= 0 there exists i ∈ I such that pi(x0) 6= 0. If Ui = { x | pi(x) < 1 }, then the family of all scalar multiples λU , Tn λ > 0, where U runs through ﬁnite intersections U = j=1 Uj , forms a base of the neighbourhoods of the origin of a locally convex topology t on E in which all the seminorms pi are continuous. Each lcs (E, t) can be described in such a way. If we put in the previous construction Ui = { x | pi(x) 6 1 }, we obtain a base of neighbourhoods of the origin constituted of closed absolutely convex sets.

Corollary. A locally convex topology on a vector space is metrizable if and only if it can be described by a countable family of seminorms. The following theorem is due to Kolmogoroﬀ.

Theorem 3.3. A tvs (E, t) is normable if and only if there exists an absolutely convex and bounded neighbourhood of the origin in E . Proof. We have only to prove that the condition is suﬃcient. Let U be an aboslutely convex and bounded neighbourhood of the origin in (E, t). The Minkowski functional p = pU is obviously a norm on E . As in the proof of the Proposition 2.8 we obtain that the boundness of U implies that the family 1 { n U | n ∈ N } is a base of neighbourhoods of the origin for the topology t, and hence the norm p generates this topology. The main property which distinguishes locally convex tvs’s from the class of arbitrary tvs’s is expressed by the Hahn-Banach theorem. We shall not prove this theorem (referring to [62], [90], [103]), but we shall state the following two equivalent formulations.

Theorem 3.4. Let (E, t) be a tvs, A its nonempty open and convex subset and F a linear manifold in E (i.e. a set of the form F = x0 + F0 , where x0 ∈ E and F0 is a subspace of E ) which does not intersect A. Then there exists a closed hyperplane H disjoint with A.

Theorem 3.5. Let E be a vector space and p a seminorm on E . Let F be a vector subspace of E and f : F → K a linear functional, such that |f(x)| 6 p(x) for x ∈ F . Then there exists a linear functional f˜: E → K which agrees with f on the subspace F , and which satisﬁes |f˜(x)| 6 p(x) for x ∈ E .

Let us state some important corollaries of the given theorem.

Corollary 1. Let E be an lcs, F a subspace of E and f ∈ F 0 (i.e. f is a continuous linear functional on F ). Then there exists f˜ ∈ E0 such that f˜(x) = f(x) for x ∈ F . In particular, the set E0 of all continuous linear functionals on a (nontrivial) lcs E is nontrivial, i.e. contains elements distinct from 0.

Example. The space Lp , 0 < p < 1, has the trivial dual space, E0 = {0}. For the proof see [62]. 1.4. Projective and inductive topologies 15

Corollary 2. Let E be an lcs, F its closed subspace and x0 ∈ E \ F . Then 0 there exists f ∈ E such that f(x0) = 1 and f(x) = 0 for x ∈ F . Corollary 3. In an lcs (E, t) each closed linear manifold is the intersection of all closed hyperplanes containing this manifold.

Corollary 4. Let E be an lcs, A its closed, convex and nonempty subset 0 and x0 ∈ E \ A. Then there exist f ∈ E and α ∈ R such that f(x) > α for x ∈ A and f(x0) 6 α. Corollary 5. In an lcs (E, t) each closed convex set is the intersection of all closed halfspaces (i.e. sets of the form { x ∈ E | f(x) > α } or { x ∈ E | f(x) 6 α } for f ∈ E0 ) containing it.

1.4. PROJECTIVE AND INDUCTIVE TOPOLOGIES

Definition 1. Let E and Ei , i ∈ I be vector spaces over K, fi : E → Ei linear mappings and ti a linear (resp. locally convex) topology on Ei , i ∈ I . The projective topology on E corresponding to the family { (Ei, ti, fi) | i ∈ I } is the coarsest linear topology t on E for which all the mappings fi , i ∈ I from (E, t) to (Ei, ti) are continuous. −1 It is clear that t is the supremum of all the topologies fi (ti), i ∈ I .A base of neighbourhoods ofT the origin of the topology t can be given by all possible intersections of the form f −1(U ), where U is an arbitrary neighbourhood i∈I0 i i i of the origin in (Ei, ti), i ∈ I and I0 is a finite subset of I . It is also clear that the topology t is locally convex if all ti ’s are such. It is Hausdorff (i.e. (E, t) is a tvs, resp. lcs) if and only if for each x ∈ E there exist i ∈ I and a neighbourhoodT of the origin Ui in Ei such that fi(x) ∈/ Ui ; in other words if and only if i∈I ker(fi) = {0}. The following proposition is easily verified.

Proposition 4.1. A mapping u from a topological space F into E , where E is endowed with the projective topology corresponding to the family { (Ei, ti, fi) | i ∈ I }, is continuous if and only if for each i ∈ I the composition fi◦u: F → (Ei, ti) is continuous. We give now some of the most important examples of projective topologies. Examples. 1◦ Let (E, t) be a tvs and F its vector subspace, endowed with the induced topology t|F . Then (F, t|F ) is a tvs, and t|F is the projective topology corresponding to the family consisting of the single mapping — canonical embedding i: F → E . ◦ Q 2 Let (Ei, ti), i ∈ I be tvs’s (particularly, lcs’s), E = i∈I Ei the direct product of the sets Ei , and let t be the product topology of the topologies ti . 16 I. Topological vector spaces

Then t is the projective topology corresponding to the family { (Ei, ti, pi) | i ∈ I }, where pi : E → Ei are canonical projections. 3◦ Let an index set I be (partially) ordered by an order relation 6. Let { (Ei, ti) | i ∈ I } be a family of tvs’s (lcs’s) and gij for all i 6 j are continuousQ linear mappings from Ej into Ei . Let E be the subspace of the product i∈I Ei whose elements x = (xi)i∈I satisfy the condition xi = gij(xj) for all i 6 j . The space E is called the projective limit of the family { Ei | i ∈ I } corresponding to the mappings gij (i, j ∈ I , i 6 j ) and will be denoted as lim←− gij (Ej). The topology t on E is the projective topology corresponding to theQ family { (Ei, ti, fi) | i ∈ I }, where fi is the restriction to E of the projection pi : j∈I Ej → Ei . Another important type of projective topologies, the so called weak topologies, will be described in the paragraph 1.6. The following property will be used for constructing examples in the Chapter II.

Proposition 4.2. Each lcs (tvs) (E, t) is topologically isomorphic to a subspace E0 of a topological product of Banach (Fréchet)spaces. E is complete if and only if the subspace E0 is closed. Proof. We shall prove the proposition in the locally convex case; in the topological vector case the proof is similar. Let { pi | i ∈ I } be a system of −1 seminorms on E which defines the topology t. If Ni = pi (0), then E/Ni =: Ei is a normed space with the normp ˜i defined byp ˜i([x]i) = pi(x), where [x]i is the equivalence class of x in Ei . Let E˜i be the completion of Ei (with the norm also Q ˜ denoted byp ˜i ) and F = i∈I Ei . Then F is a product of Banach spaces and we have to show that (E, t) is isomorphic to its subset E0 .

To each x ∈ E correspond the element x0 = ([x]i)i∈I ∈ F . The mapping x 7→ x0 is obviously linear and injective; denote by E0 its image in F . It is easy to check that topology on E0 which corresponds to the topology t (under the given embedding) coincides with the product topology that F induces on E0 .

If (E, t) is complete, it follows that E0 is closed; conversely, if E0 is closed in F , then E0 is complete (F is complete since it is easy to show that arbitrary products of complete spaces are complete), and so (E, t) is complete, too. Bounded subsets in projective linear topologies can be easily described.

Proposition 4.3. 1◦ Let (E, t) be a tvs and (F, t|F ) its subspace. A subset B ⊂ F is bounded in the topology t|F if and only if it is bounded in (E, t). ◦ Q 2 If (E, t) = i∈I (Ei, ti) is the product of tvs’s (EQi, ti). A subset B ⊂ E is bounded in the product topology t if and only if B ⊂ i∈I Bi , where Bi are bounded sets in (Ei, ti), i ∈ I . ◦ Proof. 2 It can be easily seen, on the base of the deﬁnitionQ of the product topology, that if Bi are bounded subsets of Ei , i ∈ I , then i∈I Bi is bounded in E . On the other hand, if B is bounded in E , then pi(B) are boundedQ in Ei , since the projections pi : E → Ei are continuous and, besides, B ⊂ i∈I pi(B). 1.4. Projective and inductive topologies 17

Definition 2. Let E and Ei , i ∈ I are vectorS spaces over K, fi : Ei → E , i ∈ I are linear mappings such that E = span i∈I fi(Ei) and let ti be a linear (resp. locally convex) topology on Ei , i ∈ I . The ∗-inductive topology (resp. the inductive topology) on E corresponding to the family { (Ei, ti, fi) | i ∈ I } is the finest linear (resp. locally convex) topology t on E for which all the mappings fi :(Ei, ti) → (E, t), i ∈ I are continuous. In contrast to the projective topology, the concept of the inductive topology depends on the category in which it is defined — whether we are concerned with the whole category of tvs’s or only with the lcs’s; that is the reason why the different notation is used. For example, it can be seen easily that if E is an arbitrary vector space with a Hamel basis { ei | i ∈ I } and Ei = { λei | λ ∈ K } with the topology isomorphic to the usual topology of K, and if fi denotes the canonical embedding Ei into E , i ∈ I , then the ∗-inductive topology corresponding to the family { (Ei, ti, fi) | i ∈ I } is the finest linear topology tm on E , while the inductive topology is the finest locally convex topology tl . The last two topologies are different if the set I is uncountable. Observe also that the ∗-inductive (inductive) topology, in general, need not be Hausdorff, even if all ti ’s are such. Similarly to the projective case, we have Proposition 4.4. A mapping u from the space (E, t) into a topological space F , where t is the ∗-inductive (inductive) topology corresponding to the family { (Ei, ti, fi) | i ∈ I } is continuous if and only if for each i ∈ I the composition u◦fi : Ei → F is continuous. Let us describe now how neighbourhoods of the origin in inductive topologies look like. Proposition 4.5. 1◦ Let t be the ∗-inductive topology on E , corresponding to the family { (Ei, ti, fi) | i ∈ I } and let Fi be a family of ti -topological strings in Ei , i ∈ I such that all of their knots form a base of neighbourhoods of the origin i in (Ei, ti). If Ui = (Un) ∈ Fi and if a string U = (Un) in E is constructed as X∞ ½[ ¾ ¡ i ¢ (∗) Un = fi U2n−1k , k=1 i∈I then all such strings are t-topological and the family of all the knots of all such strings forms a base of neighbourhoods of the origin for (E, t).

If the index set is countable, I = N, then the sets Un , instead in the form (∗), can be taken as X∞ i Un = fi(Un). i=1 2◦ Let t be the inductive topology on E , corresponding to the family of locally convex spaces and linear mappings { (Ei, ti, fi) | i ∈ I }. A base of neighbourhoods 18 I. Topological vector spaces of the origin for the topology t is formed by the family of all absolutely convex and −1 absorbing subsets U of E , such that for each i ∈ I , the inverse image fi (U) is a neighbourhood of the originS in (Ei, ti). This base can be created by the sets of the form Γi∈I fi(Ui) = Γ i∈I fi(Ui), where Ui runs through a base of absolutely convex neighbourhoods of the origin in (Ei, ti), i ∈ I . ◦ Proof. 1 To prove that all of the strings U = (Un) are t-topological, it is −1 suﬃcient to prove that for each i ∈ I and each n ∈ N, the set fi (Un) is a neighbourhood of the origin in (Ei, ti). But this follows from

µX∞ ½[ ³ ´¾¶ −1 −1 j i fi (Un) = fi fj U2n−1k ⊃ U2n−1k. k=1 j∈I

−1 Let V = (Vn) be an arbitrary t-topological string. Then fi (V) is a ti - i topological string, i ∈ I , and so there exist strings Ui = (Un) ∈ Fi , such that i −1 S i Un ⊂ fi (Vn+1) for i ∈ I , n ∈ N. It follows that i∈I fi(Un) ⊂ Vn+1 and if we P∞ deﬁne Un as in (∗) we have Un ⊂ V2n−1k+1 ⊂ Vn . Hence U = (Un) ⊂ V , which k=1 means that the knots of the strings of the form (∗) create a base of neighbourhoods of the origin for the topology t.

In the countable case, if V = (Vn) is a t-topological string, one can ﬁnd strings i i −1 Ui = (Un) ∈ Fi , i ∈ N such that Un ⊂ fi (Vn+i). Then one obtains

X∞ X∞ i Un := fi(Un) ⊂ Vn+i ⊂ Vn, i=1 i=1 which means that U = (Un) ⊂ V . 2◦ follows immediately from the definition. Examples. 1◦ Let (E, t) be a tvs, F a vector subspace of E and q : E → E/F the quotient mapping, given by q(x) = [x] = x + F for x ∈ E . The quotient- topology t/F is defined as the finest topology on E/F for which the mapping q is continuous. Open sets in t/F are the sets of the form q(G), where G are open sets in E , i.e. the mapping q is open, too, and it maps a base of neighbourhoods of the origin from (E, t) onto a base of neighbourhoods of the origin in (E/F, t/F ). The topology t/F is Hausdorff if and only if F is a closed subspace of E . It is locally convex if and only if t is locally convex. It is clear that t/F is the ∗-inductive topology (resp. inductive topology) for the (one-element) family {(E, t, q)}. ◦ 2 Let, for each i ∈ I ,(Ei, ti) be a tvs and denote by E the algebraic direct L ∗ sum of the spaces Ei , i.e. E := i∈I Ei . Let t be the ∗-inductive topology on E corresponding to the family { (Ei, ti, ji) | i ∈ I }, where ji : Ei → E are canonical embeddings, i.e. the finest linear topology such that all these embeddings ∗ are continuous. The space (E, t ) is called the ∗-direct sum of the spaces (Ei, ti) 1.4. Projective and inductive topologies 19

L and is denoted by ∗ − i∈I (Ei, ti). SinceQ this topology is obviuosly finer than the topology which the product-topology of i∈I Ei induces on E , it is Hausdorff. If all the spaces Ei are locally convex, the finest locally convex topology on E , such that all embeddings ji are continuous, is the inductive topology forL the family { (Ei, ti, ji) | i ∈ I }. The corresponding space is denoted by (E, t) = i∈I (Ei, ti) and called the direct sum of (locally convex) spaces (Ei, ti). In general, the topology t is coarser then t∗ . Its base of neighbourhoods of the origin can be created by the sets of the form ½ ¾ P P U = Γ ji(Ui) = λiji(xi) | |λi| 6 1, xi ∈ Ui , i∈I where Ui runs through a base of neighbourhoods of the origin in (Ei, ti), i ∈ I . It follows form the Proposition 4.4 that the topologies t and t∗ coincide if and only if the index set I is countable, which means that in this case the ∗-direct sum of locally convex spaces is locally convex. Using the Proposition 4.3 it can be easily deduced that each (∗-) inductive topology can be represented as a quotient of the (∗-) direct sum of lcs’s (tvs’s). ◦ 3 Let E be a vectorS space and { Ei | i ∈ I } a family of distinct subspaces of E , such that E = i∈I Ei . Let each Ei be equipped with a Hausdorff locally convex (linear) topology ti so that if Ei ⊂ Ej , then tj|Ei 6 ti . If fi : Ei → E are canonical embeddings, then the space E equipped with the (∗-) inductive topology corresponding to the family { (Ei, ti, fi) | i ∈ I } is called the (∗-) inductive limit of the spaces (Ei, ti) and is denoted by (E, t) = lim−→(Ei, ti) (resp. (E, t) = ∗ − −→lim(Ei, ti)). If, moreover, tj|Ei = ti whenever Ei ⊂ Ej , the (∗-) inductive limit is called strict. The (∗-) direct sum of a family of spacesL (Ei, ti), i ∈ I is one of the examples of (∗-) inductive limits. Namely,L (∗-) i∈I (Ei, ti) can be represented as (∗-) lim−→(FJ , τJ ), where (FJ , τJ ) = i∈J (Ei, ti) and J runs through all nonempty finite subsets of the index set I . In the sequel we shall give more examples of inductive limits, some of which are important in the theory of distributions. 4◦ Let X be a Hausdorff locally compact topological space and K(X) the vector space of all continuous functions f : X → K having a compact support, with the usual linear operations. For each compact A ⊂ X denote by KA(X) the vector subspace of the space K(X) containing all functions whose supports are in A. Let U be the set of all absolutely convex and absorbing sets U from K(X), such that U ∩ KA(X), for each A, is a neighbourhood of the origin in KA(X) in the topology of uniform convergence tA on A (i.e. in the topology generated by the norm kfk = supx∈A |f(x)|). Then U is a base of neighbourhoods of the origin for a locally convex topology on K(X) which is the finest locally convex topology on K(X) inducing on each KA(X) a topology which is coarser than the topology of uniform convergence. It is obviously the inductive limit topology for the family { (KA(X), tA, iA) | A compact in X }, where iA are natural embeddings. 5◦ Let D be the space of all real (or complex) infinitely differentiable functions with compact support in (−∞, ∞). Denote by Dn the vector subspace of D 20 I. Topological vector spaces containing all functions from D whose supports lie in [−n, n], n ∈ N and introduce a metrizable topology in it using the family of seminorms

(m) pm(f) = sup |f (x)| (m = 0, 1,... ). −n6x6n As a base of neighbourhoods of the origin in D take the family of all absorbing and absolutely convex sets in D which intersect each Dn by a neighbourhood of the origin. The topology obtained in a such a way is obviously an inductive limit topology and it is important in the theory of distributions. It is clear that a similar construction is possible for the functions deﬁned on Rk . 6◦ Let D be a domain in a complex plane and H(D) the space of all holomorphic functions on D, endowed with the topology of compact convergence ◦ (see 4 ). This topology can be deﬁned by the seminorms pn(x) = supx∈An |f(x)|, where An = { x | |x| 6 n, d(x, {D) > 1/n }.

Indeed, each An is compact and if A is a compact subset of D, then d(x, {D) > 0 and so there exists n such that An ⊂ A. It follows that H(D) is metrizable. The problem of characterizing all bounded subsets in inductive topologies is more complicated then for projective ones. It is easy to construct examples where a bounded subset in a quotient E/F is contained in no canonical image of a bounded set from E . In connection with this we introduce the following notion. We say that the quotient mapping q : E → E/F lifts bounded sets (with closure) if for each bounded set B from E/F there exists a bounded set A in E such that q(A) ⊃ B (resp. q(A) ⊃ B ). This notion will be important in the so called “three-space- problem”.

Proposition 4.6. A subset B of the (∗-) direct sum of lcs’s (tvs’s) Ei , i ∈ I is bounded if and only if there exists a finite subset J ⊂ I , such that pi(B) = {0} Lfor i∈ / J and pi(B) is bounded in Ei for i ∈ J , where pi is the projection of i∈I Ei onto Ei . Proof. We shall prove the assertion in the locally convex case; in the linear topological case it is similar.L The given condition is clearly sufficient for the boundness of a set B ⊂ i∈I Ei — let us prove that it is also necessary. If the set B is bounded, since the projections pi are continuous, all the sets pi(B) are bounded in Ei , i ∈ I . Suppose that there is an infinite (e.g. countable) 0 set J = { in | n ∈ N } ⊂ I , such that pin (B) 6= {0}, n ∈ N. Then there exsits a sequence (y(n)) ⊂ B such that y(n) 6= 0, n ∈ N. Moreover, as the spaces E are in in Hausdorff, yn ∈/ nVn , where Vn is a suitable circled neighbourhood of the origin in

Ein , n ∈ N. If we denote by U the neighbourhood of the origin U := Γi∈I ji(Ui) L 1 (n) in i∈I Ei , such that Uin = Vn for all n, then n y ∈/ U for some n ∈ N, which contradicts the boundness of the set B . For bounded subsets in strict (∗-) inductive limits, the similar characterization need not be true. Without a proof we state the following 1.5. Topologies of uniform convergence 21

Proposition 4.7. Let (E, t) = lim−→(En, tn) be a strict inductive limit of a sequence of locally convex spaces and let for each n, En be a closed subspace of (En+1, tn+1). Then a set B ⊂ (E, t) is bounded if and only if it is contained and bounded in some (En, tn). An analogous assertion is valid in the topological vector case.

1.5. TOPOLOGIES OF UNIFORM CONVERGENCE. THE BANACH-STEINHAUS THEOREM

Let (E, t) and (F, s) be two tvs’s. By L(E,F ) we shall denote the vector space of all continuous linear mappings from E to F . Let M be a family of subsets of the space E . We deﬁne the topology on L(E,F ) of the uniform convergence on elements of M in the following way. Let V be a neighbourhood of the origin in F and M ∈ M; by T (M,V ) denote the set

T (M,V ) := { f ∈ L(E,F ) | f(M) ⊂ V }.

The family of all T (M,V ), when M runs through M and V runs through a base U of neighbourhoods of the origin in F , forms a base of neighbourhoods of the origin for a topology on L(E,F ) which we shall call the M-topology.

Proposition 5.1. An M-topology on L(E,F ) is linear if and only if the set f(M) is bounded in F for all f ∈ L(E,F ) and M ∈ M. If, moreover, F is an Slcs, M-topology is locally convex. If all the sets M ∈ M are bounded in E and if M∈M M is total in E , then the M-topology on L(E,F ) is Hausdorff. Proof. We can suppose that the base U of neighbourhoods of the origin in F consists of circled subsets. Since for each λ 6= 0, T (M, λV ) = λT (M,V ), the sets T (M,V ) are in this case circled. It can be easily checked that if U and V are neighbourhoods of the origin in F , such that V + V ⊂ U , then T (M,V ) + T (M,V ) ⊂ T (M,U). Thus, for the proof of the first part of the proposition it is sufficient to show that the sets T (M,V ) are absorbing if and only if for each f ∈ L(E,F ) and each M ∈ M the set f(M) is bounded in F . This, however, follows from the fact that for each such f , M and for each V ∈ U , λ > 0, the condition f(M) ⊂ λV is equivalent to the condition f ∈ T (M, λV ) = λT (M,V ). If the base U of neighbourhoods of the origin in F consists of convex sets V , then the sets T (M,V ) are also convex, which proves the second assertion. S Finally, suppose that the sets M ∈ M are bounded in E and that M∈M M is total in E . Let f ∈ L(E,F ), f 6= 0. Then there exists x0 ∈ M0 ∈ M such that f(x0) 6= 0. Since F is a Hausdorff space, for some V0 ∈ U we have f(x0) ∈/ V0 . From here it follows that f∈ / T (M0,V0), which means that the M-topology on L(E,F ) is Hausdorff. 22 I. Topological vector spaces

The space L(E,F ) equipped with the M-topology will be denoted by LM(E,F ). Examples. 1◦ Let M be the family of all ﬁnite subsets of E . In this case the M-topology will be called the topology of simple convergence, and the space LM(E,F ) will be denoted by Ls(E,F ). 2◦ If M is the family of all compact (precompact) subsets of E , the M- topology on L(E,F ) will be called the topology of compact (precompact) convergence, and the corresponding space will be denoted by Lc(E,F )(Lp(E,F )). 3◦ Let M be the family of all bounded subsets of E . We shall call this M-topology the topology of uniform convergence on bounded sets or the strong topology. The space of continuous linear functions equipped with this topology will be denoted by Lb(E,F ). In particular, if E and F are normed spaces, then the strong topology is the usual norm-topology given by kfk = supkxk=1 kf(x)k for f ∈ L(E,F ). According to the Proposition 5.1, M-topologies are usually considered when E is an lcs, and M is a family of bounded sets in it, M ⊂ B(E). We can expect more regular behaviour of such a topology if the family M satisﬁes some additional conditions. We shall say that it is saturated if 1◦ M ∈ M, N ⊂ M =⇒ N ∈ M; 2◦ M ∈ M, λ ∈ K =⇒ λM ∈ M; ◦ 3 M1,M2 ∈ M =⇒ Γ(M1 ∪ M2) ∈ M. It can be easily seen that for each family M of bounded subsets there exists the smallest saturated family M containing it — we shall call this family the saturated cover of M. It is clear, too, that the topologies tM and tM are identical. These are the reasons that in the sequel we shall usually consider only saturated families. Now we give conditions that a set of continuous linear mappings is bounded in an M-topology (M-bounded). Proposition 5.2. Let H be a subset of L(E,F ). Then the following conditions are equivalent. 1◦ H is bounded in the M-topology. ◦ T −1 2 For each neighbourhood of the origin V in F , the intersection f∈H f (V ) absorbs all the sets M ∈ M. ◦ S 3 For each M ∈ M, the union f∈H f(M) is bounded in F . Proof. 1◦ =⇒ 2◦ . We can suppose that the neighbourhoods V are circled. If the set H is M-bounded, then it is absorbed by each T (M,V ), M ∈ M, V a neighbourhood of the origin in F . It follows that f(M) ⊂ λV for each f ∈ H and T −1 some λ > 0, wherefrom it follows that M ⊂ λ f∈H f (V ). ◦ ◦ 2 =⇒ 3 . If M ∈ M and aT circled neighbourhood of the origin V in F are given, then the inclusion M ⊂ λ f −1(V ) implies the inclusion f(M) ⊂ λV f∈H S for each f ∈ H . It follows that the set f∈H f(M) is bounded in F . 1.5. Topologies of uniform convergence 23

3◦ =⇒ 1◦ . For the given M and V there exists such a λ > 0 that if for each f ∈ H , f(M) ⊂ λV is valid, then H ⊂ λT (M,V ). It follows that H is bounded in the M-topology. A special type of subsets of L(E,F ), bounded in all M-topologies, are equicontinuous subsets.

Definition 1. A subset H of the space L(E,F ) is called equicontinuous if for each neighbourhood of the origin V in F there exists a neighbourhood of the origin U in E such that f(U) ⊂ V for all f ∈ H . Similarly to the previous proposition we can prove

Proposition 5.3. For a subset H of the space L(E,F ) the following conditions are equivalent. 1◦ H is equicontinuous. ◦ T −1 2 For each neighbourhood of the origin V in F , the intersection f∈H f (V ) is a neighbourhood of the origin in E . ◦ 3 For each neighbourhood of theS origin V in F there exists a neighbourhood of the origin U in E such that f∈H f(U) ⊂ V . Corollary. If all the sets M ∈ M are bounded in E , then an equicontinuous set H ⊂ L(E,F ) is bounded in the M-topology on L(E,F ).

The converse of the previous corollary is not valid. In order to ﬁnd some suﬃ- cient conditions which ensure that each simply bounded subset of linear mappings is equicontinuous, we shall introduce some important classes of spaces. By a barrel in an lcs (E, t) we shall understand each closed, absolutely convex and absorbing set. Ultrabarrel in a tvs (E, t) is a closed string. It is clear that each lcs (tvs) possesses a base of neighbourhoods of the origin created of barrels (a base of topological strings created of ultrabarrels), but in general not each barrel (ultrabarrel) in an lcs (tvs) is a neighbourhood of the origin (topological string). Spaces in which the last is true anyway, are given the special name.

Definition 2. An lcs (tvs) is called barrelled (ultrabarrelled) if each barrel (ultrabarrel) in it is a neighbourhood of the origin (topological string). It is clear that each ultrabarrelled space which is in the same time an lcs, is also barrelled. That the converse is not true (i.e. that there exist barrelled lcs’s which are not ultrabarrelled) will be shown later. Examples. 1◦ Each Baire tvs (E, t) is ultrabarrelled (and so also barrelled if it is an lcs). In particular, each complete metrizable tvs is ultrabarrelled. S∞ In fact, if U = (Un) is an ultrabarrel in E , then E = n=1 nU2 , since U2 is a circled and absorbing set. As far as E is of the second category in itself, and the sets nU2 are closed, for each n0 ∈ N the set n0U2 contains an interior point. Thus, U2 contains an interior point, say y , and, since U2 is circled, −y ∈ U2 . 24 I. Topological vector spaces

1 1 But then 0 = 2 y + (− 2 y) belongs to the interior of the set U1 , and hence U1 is a neighbourhood of the origin in (E, t). 2◦ As the previous example shows, each Banach space is ultrabarrelled. However, there exist (noncomplete) normed spaces which are not barrelled. Let E be the vector space over R created by all continuous functions x: [0, 1] → R which vanish in some (dependent on x) neighbourhood of the point t = 0. Let, furthermore, E be endowed with the topology of uniform convergence. It can be easily checked that D = { x | n|x(1/n)| 6 1, n ∈ N } is a barrel in E ; we shall show that it is not a neighbourhood of the origin.

It is suﬃcient to prove that for an arbitrary number r ∈ N there exists xr ∈ E such that kxrk 6 1/r and xr ∈/ D, which will imply that 0 is not an interior point of D. Consider two closed disjoint subsets [0, 1/4r] and [1/(r + 1), 1] of [0, 1]. On the base of the Urysohn lemma there exists a continuous real-valued function, say xr , on [0, 1] with the values in [0, 1/r] such that · ¸ · ¸ 1 1 1 x (t) = 0 for t ∈ 0, and x (t) = for t ∈ , 1 . r 4r r r r + 1 µ ¶ 1 It is clear that x ∈ E and kx k 6 1/r. On the other hand, since x = r r r r + 1 1 1 > , it follows that x ∈/ D. We conclude that E is not a barrelled lcs. r r + 1 r The next theorem is one of the variants of generalizations of the classical Banach-Steinhaus theorem. Theorem 5.4. Let E and F be lcs’s (tvs’s), such that E is barrelled (ultrabarrelled). Then each simply bounded subset H of the space L(E,F ) is equicontinuous. Proof. We shall prove the theorem in the topological vector case; in the locally convex case it is even simpler. Let E be an ultrabarrelled and F an arbitrary tvs and let V be an arbitrary closed topological string in F . Then, since H is simply bounded, U = T −1 f∈H f (V) is a closed string (i.e. an ultrabarrel) in E (see Proposition 5.2). This means that U is a topological string in E . But then, using the Proposition 5.3, we conclude that H is an equicontinuous subset. Observe that the property of barrelled (ultrabarrelled) spaces given in this theorem is characteristic for them, i.e. each lcs (tvs) E which, for every lcs (tvs) F possesses the given property [it is enough to consider only complete metrizable F ’s], is barrelled (ultrabarrelled). The proof is trivial in the locally convex case (see Proposition 6.4); for the proof in the topological vector case see [126]. In order to give a new form to this theorem, we shall deduce now some more properties of equicontinuous sets. Observe ﬁrst of all that the topology of simple convergence can be introduced also on the set F E of all functions from E into F (it coincides there with the product topology). 1.5. Topologies of uniform convergence 25

Lemma 5.5. Let H ⊂ L(E,F ) be equicontinuous and let H1 be its closure in E F equipped with the topology of simple convergence. Then H1 is also contained in L(E,F ) and it is equicontinuous, too.

Proof. If f1 ∈ H1 , it can be easily proved that the mapping f1 is linear. Since H is equicontinuous, there exists a neighbourhood of the origin U in E such that for all f ∈ H we have f(U) ⊂ V , where V is the given closed neighbourhood of the origin in F . The continuity of the mapping f 7→ f(x) of the space F E to F implies that f1(x) ∈ V for all f1 ∈ H1 , x ∈ U . Thus, H1 is equicontinuous in L(E,F ).

Lemma 5.6. Let H be an equicontinuous subset of L(E,F ). Then the restrictions of the topology of simple convergence and the topology of precompact convergence coincide on H . Proof. We have only to prove that the restriction on H of the topology of precompact convergence is coarser than the restriction of the topology of simple convergence. In other words, we have to prove that for each f0 ∈ H , each neighbourhood of the origin V in F and each precompact subset P ⊂ E there exist a ﬁnite subset P0 ⊂ E and a neighbourhood of the origin V0 in F such that

{f0 + T (P0,V0)} ∩ H ⊂ f0 + T (P,V ).

Choose, first of all, a neighbourhood of the origin W in F such that W +W + W ⊂ V and a circled neighbourhood of the origin U in E such that h(U) ⊂ W for each h ∈ H . Using the precompactness of the set P , we can find a finite set P0 = { xi | i = 1, . . . , m } ⊂ P , such that P ⊂ P0 + U . Choose now a circled Pm neighbourhood of the origin V0 in F , such that that V0 ⊂ W . If g ∈ T (P0,V0), i=1 then g(xi) ∈ V0 for each i and

Pm g(P ) ⊂ V0 + f(U) ⊂ W + g(U). i=1

Let now f0 ∈ H and h ∈ H ∩ {f0 + T (P0,V0)}. Then h = f0 + g , where g ∈ T (P0,V0). Since g = h−f0 , g(U) ⊂ h(U)+f0(U) ⊂ W +W , because U is circled. Thus, g(P ) ⊂ V , h = f0 + g ∈ f0 + T (P,V ) and the proof is completed. The following is a stronger version of the Banach-Steinhaus theorem.

Theorem 5.7. Let E and F be tvs’s (lcs’s) such that E is ultrabarrelled (barrelled). If a ﬁlter F in L(E,F ) is bounded in the topology of simple convergence (i.e. if it contains a set bounded in this topology) and if it converges to a function E f1 ∈ F in this topology, then f1 ∈ L(E,F ) and F → f1 uniformly on each precompact subset of the space E . Proof. Let H ∈ F be bounded in the topology of simple convergence on L(E,F ). On the base of the Theorem 5.4, H is equicontinuous. If H1 denotes 26 I. Topological vector spaces

E the closure of H in F , then f1 ∈ H ; by the Lemma 5.5, the set H1 is contained in L(E,F ) and it is equicontinuous. Since, by the Lemma 5.6, the topologies of simple and precompact convergence are the same on H1 , the theorem is proved. We give now the promised example of a barrelled lcs which is not ultrabarrelled. Example. Let (E, t) be the sequence space l1/2 . It is a non-locally convex, complete and metrizable tvs, hence it is ultrabarrelled. It can be easily checked that, if t◦◦ denotes the locally convex topology associated to t (i.e. the ﬁnest locally convex topology coarser then t), then the space (E, t◦◦) is barrelled. Let us show that it is not ultrabarrelled.

Consider mappings fn : E → E given by fn(x1, x2,... ) = (x1, . . . , xn, 0,... ). ◦◦ fn ’s are continuous linear mappings from (E, t ) onto (E, t) and for each x ∈ E , ◦◦ fn(x) → x in t. The set { fn | n ∈ N } ∪ {i}, where i:(E, t ) → (E, t) is the identical mapping, is simply bounded but not equicontinuous, since i is not continuous. By the Theorem 5.4, (E, t◦◦) is not an ultrabarrelled space. In order to make variations of the Theorem 5.4, in such a sense that for the equicontinuity of the set H boundness in some other topologies are required, we shall introduce now some new classes of tvs’s (lcs’s). A set V in a tvs E is called bornivorous if it absorbs all bounded subsets of E .

Definition 3. 1◦ An lcs (tvs) in which each bornivorous barrel (bornivorous ultrabarrel) is a neighbourhood of the origin (topological string) is called quasibarrelled (quasiultrabarrelled). 2◦ An lcs (tvs) in which each bornivorous absolutely convex set (bornivorous string) is a neighbourhood of the origin (topological string) is called bornological (ultrabornological). The relationships quasibarrelled–quasiultrabarrelled and bornological–ultrabornological (locally convex) spaces are similar as in the barrelled case. Further- more, it is clear that each barrelled (ultrabarrelled), and also each bornological (ultrabornological) lcs (tvs) is quasibarrelled (quasiultrabarrelled), and none of these implications can be reversed. For the ﬁrst of them, an example can be constructed using the following two examples. Examples. 1◦ Each metrizable lcs (tvs) is bornological (ultrabornological) and, a fortiori, quasibarrelled (quasiultrabarrelled).

Indeed, let, e.g., E be a metrizable lcs with a base { Vn | n ∈ N } of neighbourhoods of the origin, which can be assumed decreasing. Let U be an absolutely convex bornivorous subset in E . Then for some n, Vn ⊂ nU holds. For, if this were not the case, there would exist elements xn ∈ Vn such that xn ∈/ nU . Then (xn) would be a zero-sequence in E and it would be bounded and so absorbed by the set U , which is impossible. 2◦ Let ϕ be the vector space of all real sequences having at most ﬁnitely many terms distinct from zero, endowed with the sup-norm. By the previous example, 1.5. Topologies of uniform convergence 27 it is (ultra)bornological, and so quasi(ultra)barrelled. Let us show that it is not (ultra)barrelled.

For each n ∈ N, let fn be the continuous linear functional on ϕ, deﬁned by fn(x) = nxn . Then the set { fn | n ∈ N } is simply bounded, but not equicontinuous. By the Theorem 5.4, ϕ is not a barrelled lcs. Furthermore, not each (ultra)barrelled space has to be (ultra)bornological. The ﬁrst examples of this kind were constructed by L. Nachbin and T. Shirota (see [62]). Using such examples, it is easy to construct spaces which are quasibarrelled, but neither barrelled nor bornological. Example. Let E be a barrelled space which is not bornological and F a bornological space which is not barrelled. Then E ×F is a quasibarrelled space (as a product of such spaces, which can be easily checked). However, if A is a barrel in F which is not a neighbourhood of the origin, then E × A is a barrel in E × F which is not a neighbourhood of the origin. Similarly, if B is an absolutely convex bornivorous set in E which is not a neighbourhood of the origin, then B × F has the same properties in E × F . A similar example can be constructed for tvs’s. We give now the announced version of the Banach-Steinhaus theorem for quasi(ultra)barrelled and (ultra)bornological spaces.

Theorem 5.8. 1◦ Let E and F be lcs’s (tvs’s), such that E is quasi(ultra)barrelled and let H ⊂ L(E,F ) is bounded in the strong topology on L(E,F ). Then H is equicontinuous. 2◦ Let E and F be lcs’s (tvs’s), such that E is (ultra)bornological and let H be a set of linear mappings from E into F . If H is uniformly bounded on bounded subsets of E , then it is equicontinuous. The proof of this theorem is similar to the proof of the Theorem 5.4, and so is omitted. A similar remark as the one given after this theorem is valid, too. Another important property of the introduced classes of spaces is that they are stable in connection with the (∗-) inductive topology.

Proposition 5.9. If spaces (Ei, ti), i ∈ I are (ultra)barrelled, quasi(ultra)- barrelled or (ultra)bornological, and if t is the (∗-) inductive topology on the vector space E corresponding to the family { (Ei, ti, ji) | i ∈ I }, then the space (E, t) is of the respective type. In particular, all the mentioned classes are stable when passing to the quotient, (∗-) direct sum and (∗-) inductive limit. Proof. Let us prove this proposition for barrelled spaces; for other classes the proof is similar.

Let all the spaces (Ei, ti) be barrelled and let U be a barrel in (E, t). It can −1 be easily checked that for each i ∈ I , Ui = ji (U) is a barrel in (Ei, ti), and so a neighbourhood of the origin in that space. It follows that U is a neighbourhood of the origin in (E, t), and so this space is barrelled. 28 I. Topological vector spaces

Bornological spaces have an important property which can be considered as a kind of converse of the previous proposition.

Proposition 5.10. Each bornological space can be represented as an inductive limit of a family of normed spaces. Proof. Let (E, t) be a bornological locally convex space and let B be the family of all closed, boundedS and absolutely convex subsets of E . For each B ∈ B consider the space EB = n∈N nB , equipped with the norm pB which is the Minkowski functional of the set B . It is obvious that the topology of the space (EB, pB) is finer then the one induced on EB by the topology t of the given space. If iB are natural embeddings of EB into E , the inductive topology t0 on E , corresponding to the family { (EB, pB, iB) | B ∈ B }, is obviously the finest topology on E having the same bounded sets as t. Since the space (E, t) is bornological, the topologies t0 and t coincide, and the proposition is proved. For the corresponding proposition for tvs’s see [3]. We shall finish this paragraph stating, without proofs, some of many variants of the closed graph and open mapping theorems, which, together with the Hahn- Banach and Banach-Steinhaus theorems, represent three fundamental principles of linear analysis. We shall choose variants which are closest to the concepts we have treated in this paragraph, referring to [63], [90], [103] for other possible variants. A mapping from a tvs (E, t) into another tvs (F, u) is said to be closed if its graph is closed in (E, t) × (F, u). Theorem 5.11. A tvs (lcs) (E, t) is ultrabarrelled (barrelled) if and only if each closed linear mapping from (E, t) into an arbitrary complete metrizable tvs (lcs) is continuous. Theorem 5.12. Let E and F be complete metrizable tvs’s and f : E → F a continuous linear mapping whose image f(E) is dense in F . Then either f(E) is of the first category in F , or f(E) = F and f is a topological homeomorphism (i.e. an open mapping.) Corollary. A continuous linear mapping f of a complete metrizable tvs E into another space F of the same kind is a topological homeomorphism if and only if f(E) is closed in F . 1.6. Duality theory 29

1.6. DUALITY THEORY

Let F and G be two vector spaces over K and let f be a bilinear form on F × G satisfying the following two conditions: ◦ 1 if f(x0, y) = 0 for all y ∈ G, then x0 = 0; ◦ 2 if f(x, y0) = 0 for all x ∈ F , then y0 = 0. Then the triple (F, G, f) is called a dual system or a duality over K. We also say that f establishes a duality between F and G; it is called the canonical bilinear form of the given duality. The value of this form is also denoted as f(x, y) = hx, yi, and the duality itself as hF,Gi (if its form f is understood). The following two are the most important examples of dualities. Examples. 1◦ Let E be a vector space and E∗ its algebraic dual. Then the bilinear form (x, x∗) 7→ x∗(x) = hx, x∗i defines the duality hE,E∗i. 2◦ Let E be an lcs and E0 its topological dual. Then E0 is a subspace of E∗ , and the form (x, x0) 7→ x0(x) = hx, x0i possesses the properties 1◦ and 2◦ on the base of the Hahn-Banach theorem, and so it defines the duality hE,E0i. Let hF,Gi be an arbitrary duality. Then for each y ∈ G the mapping x 7→ ∗ hx, yi represents a linear functional fy on F (i.e. an element of F ). Since the ◦ mapping y 7→ fy is linear and, by 2 , injective, it is an isomorphism of the vector space G and a certain subspace of the algebraic dual F ∗ . The canonical bilinear form of the duality hF,Gi is then the restriction on F ×G of the canonical bilinear form of the duality hF,F ∗i. We shall see in the sequel that, using suitable topologies, the previous construction can be done in the topological sense, too. Let hF,Gi be a dual pair and treat G, as just mentioned, as a subspace of the dual F ∗ . The weak topology σ(F,G) on F is the coarsest topology on F for which all the linear forms x 7→ hx, yi, y ∈ G are continuous. This topology is obviuosly defined by the family of seminorms py(x) = |hx, yi|, y ∈ G; in other words, its base of (closed) neighbourhoods of the origin can be created by the sets of the form

{ x | sup |hx, yii| 6 1, yi ∈ G, i = 1 . . . , n }. 16i6n

The topology σ(F,G) is obviously locally convex and Hausdorff. It is also the projective topology (in the sense of the paragraph 1.4) for the family { (Ey, ty, fy) | y ∈ G }, where for each y ∈ G,(Ey, ty) is the scalar field with the usual topology, and fy is the mapping x 7→ hx, yi. The following proposition can be easily verified.

Proposition 6.1. The (topological) dual space of the space (F, σ(F,G)) is equal to G, i.e. a linear form f on F is σ(F,G)-continuous if and only if it has the form f(x) = hx, yi for some (uniquely determined) vector y ∈ G. 30 I. Topological vector spaces

Of course, in the similar way the weak topology σ(G, F ) on G is deﬁned, and then (G, σ(G, F ))0 = F . If E is an lcs and hE,E0i is the corresponding dual pair (example 2◦ ), then σ(E0,E) is obviuosly the topology of simple convergence on E0 = L(E, K), in the sense of the previous paragraph. In this situation we have

Proposition 6.2. If (E, t) is an lcs, then the σ(E,E0)-closure of an arbitrary convex susbet of E coincides with its t-closure. Proof. Since t > σ(E,E0), each σ(E,E0)-closed set is also t-closed. Converse- ly, each convex t-closed subset of E is weakly closed, because it is, by a corollary of the Hahn-Banach theorem, the intersection of all weakly closed halfspaces of the form { x | f(x) 6 α }. An important auxiliary construction in the duality theory is given by the notion of (absolute) polar. It is a concrete version of the construction of sets of the form T (M,V ) from 1.5. Let hF,Gi be a dual pair. The polar of a set A ⊂ F is the set

A◦ = { y ∈ G | |hx, yi| 6 1 for all x ∈ A }.

In the similar way the polar B◦ of a set B ⊂ G is deﬁned. The following properties of polars can be easily proved: 1◦ The polar A◦ of an arbitrary set A ⊂ F is an absolutely convex and σ(G, F )-closed set. 2◦ If A ⊂ B , then B◦ ⊂ A◦ . 3◦ If λ 6= 0, then (λA)◦ = λ−1A◦ . ¡ ¢ ◦ S ◦ T ◦ 4 If { Ai | i ∈ I } is a family of subsets of F , then i∈I Ai = i∈I Ai . 5◦ If E is a tvs, then a subset B of its topological dual E0 is equicontinuous if and only if B ⊂ U ◦ for some neighbourhood of the origin U in E . The following important assertion is usually called the bipolar theorem.

Proposition 6.3. Let hF,Gi be a dual pair and A ⊂ F . Then the bipolar A◦◦ of A is equal to the σ(F,G)-closed absolutely convex cover of the set A. Proof. Denote by B the σ(F,G)-closed absolutely convex cover of the set A. Using the property 1◦ of polars, we conclude that A◦◦ is a σ(F,G)-closed and absolutely convex subset of F , obviously containing A, and so B ⊂ A◦◦ . Con- versely, suppose that x0 ∈/ B . On the base of the Hahn-Banach theorem (Corollary 4 of Theorem 3.5), there exists a σ(F,G)-continuous linear functional f , such that ◦ hx0, fi > 1 and |hx, fi| 6 1 for all x ∈ B . Since A ⊂ B and so f ∈ A , this shows ◦◦ ◦◦ ◦◦ that x0 ∈/ A . Thus, A ⊂ B and so A = B . The following corollaries can be easily deduced from the bipolar theorem. 1.6. Duality theory 31

Corollary 1. For each set A ⊂ F , A◦◦◦ = A◦ holds.

Corollary 2. If Ai , i ∈ I are σ(F,G)-closed absolutely convex sets in F , ¡T ¢◦ then i∈I Ai is equal to the σ(G, F )-closed absolutely convex cover of the set S ◦ i∈I Ai . Corollary 3. If (E, t) is an lcs, then the polars (corresponding to hE,E0i) of an arbitrary fundamental system of equicontinuous subsets of E0 form a base of neighbourhoods of the origin in (E, t). Proof. Let M be a fundamental system of equicontinuous subsets of E0 and let U be a closed absolutely convex neighbourhood of the origin in E . From the Proposition 6.2 it follows that the neighbourhood U is σ(E,E0)-closed. Since U ◦ is an equicontinuous set in E0 , there exists M ∈ M such that U ◦ ⊂ M . Hence M ◦ ⊂ U ◦◦ = U , which proves the corollary. In other words: the topology t of a tvs (E, t) is locally convex if and only if it is the topology of uniform convergence on equicontinuous subsets of E0 . The following is a dual characterization of barrelled spaces.

Proposition 6.4. Let E be an lcs. Then the family of all barrells in E and the family of all σ(E0,E)-closed and bounded absolutely convex sets in E0 correspond to each other under the mapping A 7→ A◦ , in connection with the duality hE,E0i. In other words, the space E is barrelled if and only if each σ(E0,E)- bounded subset of E0 is equicontinuous. Proof. Let U be a barrel in E . Then U ◦ is a σ(E0,E)-bounded set. On the base of the Proposition 6.2 we conclude that U is also σ(E,E0)-closed, and so by the bipolar theorem it follows that U = U ◦◦ . Hence, to finish the proof it is sufficient to show that for each σ(E0,E)-bounded and σ(E0,E)-closed absolutely convex set B ⊂ E0 , its polar B◦ is a barrel in E . The only thing which has to be checked is that B◦ is an absorbing set. Let x ∈ E . Then {x}◦ is a σ(E0,E)-neighbourhood of the origin. It follows that there exists λ > 0 such that B ⊂ λ−1{x}◦ = {λx}◦ . It follows that λx ∈ B◦ , and the proof is complete. If a dual pair hF,Gi has been given, till now we have mostly considered the weak topologies σ(F,G) and σ(G, F ). However, a lot of other topologies on the spaces F and G can be considered, particularly topologies of uniform convergence. For example, if M is a certain family of σ(G, F )-bounded subsets of the space G, the topology of M-convergence can be considered on the space F = L((G, σ(G, F )), K) (the weak topology σ(F,G) will be obtained if M is the family of finite subsets of G). A natural question is when the topological dual of the space (F, tM) will be equal to G — it is clear that σ(F,G) is the coarsest of the topologies with this property, and so we have to look for the finest one. For topologies t which have the property that (F, t)0 = G we shall say that they are compatible with respect to the dual pair hF,Gi. Let us state a common property of all compatible topologies, which can be proved in the same way as the Proposition 6.2. 32 I. Topological vector spaces

Proposition 6.5. The closure of a convex subset of F is the same in all compatible topologies on F and it coincides with its σ(F,G)-closure.

Before answering the question on determining of all the compatible topologies, we shall prove the following important theorem.

Theorem 6.6. (L. Alaoglu-N. Bourbaki) Let E be an lcs and U a neighbourhood of the origin in E . Then the polar U ◦ is a σ(E0,E)-compact set in E0 . As a consequence, each equicontinuous set in E0 is relatively σ(E0,E)-compact. Proof. As we have already said, the weak topology σ(E0,E) is the topology of simple convergence on E0 = L(E, K), and so it is induced by the product topology of KE . On the base of Tychonoff theorem, in order that the subset U ◦ ⊂ KE (which is σ(E0,E)-closed) be σ(E0,E)-compact, it is sufficient that all the sets { f(x) | f ∈ U ◦ }, x ∈ E be relatively compact in K. However, for the given x ∈ E there exists λ > 0 such that λx ∈ U , and so |f(x)| 6 1/λ for all f ∈ U ◦ , which finishes the proof.

Corollary. The unit ball of the space E0 , dual to a normed space E is a σ(E0,E)-compact set.

Theorem 6.7. (G. W. Mackey-R. F. Arens) A locally convex topology t on F is compatible with the dual pair hF,Gi if and only if it is the topology of uniform convergence on a certain saturated family M (which covers G) of σ(G, F )- relatively compact subsets of G. Proof. Let the topology t on F be compatible with hF,Gi. Using the Corollary 3 of the bipolar theorem, we deduce that it is the topology of uniform convergence on the (saturated) family of all t-equicontinuous subsets of G. On the base of the previous theorem, these subsets are relatively σ(G, F )-compact, which proves that the condition of the theorem is necessary. Conversely, suppose that M is a certain saturated family of σ(G, F )-relatively compact subsets which covers G. The proposition 5.1 implies that F , equipped 0 with the M-topology is an lcs and obviously tM > σ(F,G), hence (F, tM) ⊃ G. We have to prove the reverse inclusion. Let f ∈ F ∗ , such that f is continuous in the M-topology. The polar {f}◦ (with respect to hF,F ∗i) is, by the continuity of f , a neighbourhood of the origin in the M-topology, and so it contains a set M ◦ for some M ∈ M, which is σ(G, F )- compact and can be taken to be absolutely convex (since M is saturated). That is why f ∈ M ◦◦ , where the bipolar is taken with respect to hF,F ∗i. By the bipolar theorem, it follows that M = M ◦◦ , since M , as σ(G, F )-compact is also compact and so closed in (F ∗, σ(F ∗,F )). It follows that f ∈ M ⊂ G, which completes the proof.

Corollary 1. The topology of uniform convergence on the family of all σ(G, F )-compact absolutely convex subsets of G is the ﬁnest locally convex topology on F , compatible with the duality hF,Gi. 1.6. Duality theory 33

The topology described in this Corollary is called the Mackey topology and is denoted by τ(F,G). A locally convex space E which is equipped with its Mackey topology τ(E,E0) is called a Mackey space.

Corollary 2. (G. W. Mackey) All topologies on F compatible with hF,Gi have the same families of bounded subsets.

Proof. We have only to prove that each σ(F,G)-bounded subset of F is also τ(F,G)-bounded. If a set B is σ(F,G)-bounded, its polar B◦ is a σ(G, F )-closed, absolutely convex and absorbing set, hence a barrel in (G, σ(G, F )). We shall prove that it absorbs each σ(G, F )-compact absolutely convexS set in G. Such a set M is certainly σ(G, F )-complete, and so the space GM = n∈N nM , equipped with the norm pM deﬁned by the Minkowski functional of the set M , is a Banach space, a fortiori a barrelled space. Moreover, the normed topology of this space is ﬁner then the topology on GM induced by the topology of the space (G, σ(G, F )). Thus, ◦ ◦ B ∩ GM is a barrel in the space (GM , pM ), and so B absorbs M . It follows that the set B is bounded in the topology τ(F,G).

Example. Each quasibarrelled (in particular each barrelled and each bornological) lcs (E, t) is a Mackey space. Indeed, let M be an absolutely convex σ(E0,E)-compact set. Its polar M ◦ is a τ(E,E0)-neighbourhood of the origin, and so it absorbs each τ(E,E0)- bounded, i.e. each t-bounded set B . Therefore, M ◦ is a bornivorous barrel in (E, t), and so it is a neighbourhood of the origin, which means that the set M = M ◦◦ is equicontinuous. This proves that the families of σ(E0,E)-compact (absolutely convex) and of equicontinuous subsets of E0 coincide, which is a characteristic of Mackey spaces.

In the previous example, the crucial fact was that in the dual of a quasibarrelled 0 space (E, t), the families Cσ of σ(E ,E)-compact absolutely convex subsets and E of equicontinuous absolutely convex subsets coincide. Moreover, in fact, in this case the family E coincides with the family Bβ of all absolutely convex strongly bounded subsets. Here, a set B ⊂ E0 is called strongly bounded if it is uniformly bounded on each weakly bounded subset of E . When barrelled spaces are concerned, on the base of the Proposition 6.4, the family E coincides with the family Bσ of all absolutely convex σ(E0,E)-bounded subsets. These observations lead us naturally to consider the following families of σ(E0,E)-bounded subsets of the space E0 : F – the family of all absolutely convex covers of ﬁnite subsets of E0 ; E – the family of all absolutely convex equicontinuous sets; 0 Cσ – the family of all absolutely convex σ(E ,E)-compact sets;

Bβ – the family of all absolutely convex strongly bounded sets; 0 Bσ – the family of all absolutely convex σ(E ,E)-bounded sets. 34 I. Topological vector spaces

The respective topologies of uniform convergence on E are: σ(E,E0) – the weak topology on E ; t – the given topology of the lcs E ; τ(E,E0) – the Mackey topology; β∗(E,E0) – the topology of uniform convergence on strongly bounded subsets of E0 ; β(E,E0) – the strong topology on E . In the dual way, topologies σ(E0,E), τ(E0,E), β∗(E0,E) and β(E0,E) are defined. Besides, it is clear that just defined strongly bounded sets are in fact the sets bounded in the topology β(E0,E). It is obvious, also, that the inclusions 0 0 F ⊂ E ⊂ Cσ ⊂ Bβ ⊂ Bσ , and so the inequalities σ(E,E ) 6 t 6 τ(E,E ) 6 β∗(E,E0) 6 β(E,E0) are valid. In general, each of the given inequalities can be strict. As was already said, the equality t = τ(E,E0) characterizes the Mackey spaces, while t = β∗(E,E0) characterizes the quasibarrelled and t = β(E,E0) the barrelled spaces. When there is no danger of ambiguity, we shall use shorter 0 0 0 notations as Eβ for (E , β(E ,E)) etc. We give a sufficient condition for the equality Bβ = Bσ .A tvs is said to be quasi-complete if all of its closed and bounded subsets are complete. Clearly, each complete tvs is quasi-complete, and each quasi-complete space is sequentially complete.

Proposition 6.8 If (E, t) is a sequentially complete (in particular quasi- complete) lcs, then each weakly bounded subset in E0 is strongly bounded. Proof. Observe, first of all, that every barrel T in an arbitrary lcs (E, t) absorbs each absolutely convex, bounded and sequentially complete subset. Indeed, if B is such a subset in E , then (EB, pB) (see the proof of the Proposition 5.10) is a Banach, so a barrelled space, and its topology is finer than the topology t|EB . In such a way, T ∩ EB is a barrel in (EB, pB), which means that it absorbs the set B . Let now M be an absolutely convex weakly bounded subset of E0 , where (E, t) satisfies the conditions of the proposition. Then its polar M ◦ is a barrel in E and so it absorbs each absolutely convex, bounded subset of E , since they are 0 in our case sequentially complete. This implies the equality Bβ = Bσ in E . Corollary. Each sequentially complete quasibarrelled lcs is barrelled. Of the mentioned topologies on E , σ(E,E0), t and τ(E,E0) are compatible with the duality hE,E0i, while β∗(E,E0) and β(E,E0) in general are not. In other 0 0 words, in general (Eβ) = E does not hold. We give a special name to the spaces having this property.

0 0 Definition 1. An lcs (E, t) is called semi-reﬂexive if it is (Eβ) = E . Proposition 6.9. An lcs (E, t) is semi-reﬂexive if and only if either of the following assertions holds: 1.6. Duality theory 35

(a) each β(E0,E)-continuous linear form on E0 is σ(E0,E)-continuous; (b) the space (E0, τ(E0,E)) is barrelled; (c) each t-bounded subset of E is relatively σ(E,E0)-compact; (d) E is σ(E,E0)-quasi-complete. Proof. By the definition, each semi-reflexive space satisfies the condition (a). If T is a barrel in (E0, τ(E0,E)), it is also a barrel in (E0, σ(E0,E)) by the Proposition 6.5, and so a β(E0,E)-neighbourhood of the origin by the Proposition 6.4. But, if (a) is satisfied, then the strong topology β(E0,E) is compatible with the duality hE,E0i and so β(E0,E) = τ(E0,E), hence the space (E0, τ(E0,E)) is barrelled. The implication (b) =⇒ (c) follows also from the Proposition 6.4, since 0 if Eτ is barrelled, then each t-bounded subset is equicontinuous as a subset of 0 0 E = L(Eτ , K) and so it is relatively σ(E,E )-compact. (c) =⇒ (d) is trivial since compactness implies completeness. Finally, let E be a σ(E,E0)-quasi- complete space. By the following Lemma, each t-bounded set in it is σ(E,E0)- precompact, and so also relatively σ(E,E0)-compact, since it is contained in a bounded σ(E,E0)-complete set. It means that topologies β(E0,E) and τ(E0,E) coincide, and so E is a semi-reflexive space. Lemma 6.10. If (E, t) is an lcs, then each (weakly) bounded subset A of E is weakly precompact. Proof. The collection of all sets of the form V = { x | |hx0, xi| 6 1 }, x0 ∈ E0 is a subbase of the weak topology σ(E,E0) and it is enough to prove that the set A can be covered by finitely many sets which are small of the order V . But for each x0 ∈ E0 the set hA, x0i is a bounded set of scalars and so it can be covered by finitely many sets K1 , ... , Kn with diameters less than 1. Thus, A can be 0−1 covered by the sets x (Ki), i = 1, 2, . . . , n which are small of order V . 0 0 The space (Eβ) is usually called the strong bidual of the space (E, t) and is denoted by E00 . The inclusion E00 ⊃ E is, obviously, always valid, while the (algebraic) equality E00 = E characterizes semi-reflexive spaces. It is natural to ask about some kind of topological equality. In that sense it is most natural to consider the strong topology β(E00,E0) on E00 , which is the topology of uniform convergence on the strongly (β(E0,E)-) bounded subsets of E0 . It is clear that this topology induces on the subspace E of E00 the topology β∗(E,E0). Definition 2. An lcs (E, t) is called reflexive if it is semi-reflexive and (E00, β(E00,E0)) = (E, t) holds. The previous observations imply Proposition 6.11. A semi-reflexive lcs is reflexive if and only if it is quasibarrelled. Corollary. Each reflexive lcs is barrelled. As far as normed spaces, as metrizable ones, are quasibarrelled, it is clear that among them there is no difference between semi-reflexivity and reflexivity. 36 I. Topological vector spaces

We shall ﬁnish this paragraph by considering in more details one of the most important examples of duality — the duality between subspaces and quotients. When some other important examples, as the duality between products and direct sums or the duality between inductive and projective limits, are concerned, we refer the reader to [62] and [103]. Let F be a subspace of the lcs (E, t); the topology which t induces on F will be denoted as t|F . Let i be the topological embedding i:(F, t|F ) → (E, t). The polar F ◦ is in this case a subspace of the dual E0 ,

F ◦ = { f ∈ E0 | f(x) = 0 for x ∈ F }.

On the other hand, if F 0 is the topological dual of the space (F, t|F ), a mapping n: E0 → F 0 is defined, which to every f ∈ E0 corresponds its restriction f|F ∈ F 0 . This mapping is, clearly, an (algebraic) epimorphism. Its kernel ker n is obviuosly the mentioned polar F ◦ and so an (algebraic) isomorphismn ˆ between the quotient 0 ◦ 0 0 0 ◦ E /F and F is defined, such that n =n ˆ◦q , where q : E → E /F is the quotient mapping. Therefore, at least algebraically, the spaces F 0 and E0/F ◦ can be identified. Consider now the dual situation. Let F be a closed subspace of an lcs (E, t) and t/F the corresponding quotient-topology on E/F and q : E → E/F the quotient mapping. Let g ∈ (E/F, t/F )0 . Define the mapping f ∈ E∗ with f(x) = g([x]), x ∈ E ; this mapping is continuous, f ∈ (E, t)0 , because if U˜ is a neighbourhood of the origin in E/F , such that |g([x])| < ε for [x] ∈ U˜ , then |f(x)| < ε for x ∈ q−1(U˜) = U . We shall denote the mapping g 7→ f defined in such a way by i; it is clearly an (algebraic) embedding of the space (E/F )0 into E0 . Moreover, it is an alebraic isomorphism between (E/F )0 and the subspace F ◦ of the space E0 . Namely, if g ∈ (E/F )0 , then the functional f = i(g) vanishes on F , i.e. f ∈ F ◦ . Conversely, if f ∈ F ◦ , then the equation f(x) = g([x]) defines a linear functional g on E/F . This functional is continuous, since it follows from |f(x)| < ε, x ∈ U that |g([x])| < ε, x ∈ q(U). In such a way, we have the following algebraic relations

F 0 =∼ E0/F ◦ and (E/F )0 =∼ F ◦.

The question is whether these relations hold in the topological sense. The answer will depend on topologies we consider. 0 Let M be a saturated family of σ -bounded subsets of E and tM the respective topology of uniform convergence on E . If q is the quotient mapping of E0 onto E0/F ◦ , denote by M˜ := q(M). We have:

Proposition 6.12. The topology tM|F which tM induces on F is the same ˜ ˜ as the topology tM˜ of uniform convergence on the sets M ∈ M (which are subsets of E0/F ◦ =∼ F 0 ).

Proof. If M ∈ M is an absolutely convex set, it deﬁnes the tM|F - neighbourhood of the origin M ◦ ∩ F which comprises of all y ∈ F such that 1.6. Duality theory 37

|f(y)| 6 1 for f ∈ M . But, since f(y) = [f](y), where [f] is the equivalence class 0 ◦ ◦ ◦ of y in E /F , M ∩ F is equal to (q(M)) . Conversely, each tM˜ -neighbourhood of the origin (q(M))◦ in F , where M is absolutely convex, can be written in the form M ◦ ∩ F .

Corollary. 1◦ The weak topology σ(E,E0) induces on a subspace F the topology which is equal to the weak topology σ(F,E0/F ◦). 2◦ The Mackey topology τ(E,E0) induces on a subspace F the topology which is, in general, coarser then the Mackey topology τ(F,E0/F ◦), but is compatible with the duality hF,E0/F ◦i. Proof. The first assertion follows from the fact that when M runs through all bounded finite-dimensional subsets of E0 , q(M) runs through all bounded finite- 0 ◦ dimensional subsets of E /F . To prove the second assertion, observe that if M1 and M2 are saturated families containing all absolutely convex weakly compact 0 0 ◦ subsets of E and E /F , respectively, then q(M1) ⊂ M2 , which is a consequence of the continuity of the mapping q in the topologies σ(E0,E) and σ(E0/F ◦,F ). The last conclusion is trivial, since τ(E,E0) is finer of σ(E,E0). For topologies finer then the Mackey topology, a fortiori, equalities as in 1◦ of the previous corollary need not hold. In particular, the strong topology β(E,E0) induces on a subspace F the topology which can be strictly coarser then β(F,E0/F ◦). For the equality β(E,E0)|F = β(F,E0/F ◦) to hold, it is necessary and sufficient that each bounded subset of E0/F ◦ is contained in the closure q(B) of some bounded set B from E0 , i.e. that the quotient mapping q : E0 → E0/F ◦ lifts bounded sets with closure. Interchanging E and E0 we obtain the respective results for the topologies on the dual of a quotient-space. The details are left to the reader. Consider now possible topologies of uniform convergence on quotient spaces. Let F be a closed subspace of an lcs (E, t) and M a saturated family of absolutely convex and weakly compact subsets of E0 , and M˜ the collection of all subsets from M which lie in F ◦ ⊂ E0 .

Proposition 6.13. The quotient-topology tM/F of the topology tM on E/F is equal to the topology tM˜ . 0 0 Proof. On the base of the Mackey-Arens theorem, we have (E, tM) = E . Let i:(E/F )0 → F ◦ be the canonical isomorphism. Let us show that under 0 this isomorphism, tM/F -equicontinuous subsets of (E/F ) correspond to the tM - equicontinuous subsets of E0 which lie in F ◦ , i.e. to the elements of the family 0 0 M˜ . If M is a tM/F -equicontinuous subset of (E/F ) , then |g([x])| 6 1 for 0 g ∈ M and [x] ∈ q(U), where U is an appropriate open absolutely convex tM - neighbourhood of the origin in E . Then if f = i(g) ∈ i(M 0), |f(x)| 6 1 for all x ∈ U + F . Thus i(M 0) ⊂ U ◦ ∩ F ◦ . Conversely, if M ⊂ (E/F )0 satisﬁes i(M) ⊂ U ◦ ∩ F ◦ = (U + F )◦ , then M is contained in (q(U))◦ . This completes the proof. 38 I. Topological vector spaces

Corollary. 1◦ The quotient of the weak topology σ(E,E0) by a closed subspace F is the weak topology σ(E/F, F ◦). 2◦ The quotient of the Mackey topology τ(E,E0) by a closed subspace F is the Mackey topology τ(E/F, F ◦). Proof. 1◦ follows from the fact that ﬁnite-dimensional bounded sets in F ◦ are exactly the ﬁnite-dimensional bounded sets from E0 which are contained in F ◦ . 2◦ Using the Corollary of the Proposition 6.12 (more precisely the remark after it) we conclude that the topologies σ(F ◦, E/F ) and σ(E0,E)|F ◦ are the same. That is why the absolutely convex σ(E0,E)-compact subsets which lie in F ◦ are exactly the absolutely convex σ(F ◦, E/F )-compact subsets of F ◦ , wherefrom the assertion follows. As in the case of the topologies on subspaces, the last corollary need not hold for the strong topology. Namely, the topology β(E/F, F ◦) is always coarser then β(E,E0)/F and this inequality may be strict, because β(E,E0)/F need not be compatible with the duality hE/F, F ◦i. Interchanging E and E0 , as in the case of the Proposition 6.12, we obtain the respective conclusions for topologies on duals of subspaces. II. SUBSPACES AND QUOTIENTS OF TOPOLOGICAL VECTOR SPACES

2.1. SUBSPACES OF LCS ’S BELONGING TO THE BASIC CLASSES

The most important classes of locally convex spaces we dealt with in the ﬁrst chapter were barrelled, bornological, quasibarrelled and Mackey spaces. Taking into account the characterization of bornological spaces as inductive limits of normed spaces (Proposition I.5.10) and following Bourbaki [24], we shall call an lcs ultrabornologique (not to be confused with ultrabornological tvs’s) if it is an inductive limit of Banach spaces. All these classes of spaces are in a certain sense generalizations of the class of Banach spaces. The relationship between them can be given by the following diagram

ultrabornologique −−−−→ bornological     y y barrelled −−−−→ quasibarrelled   y Mackey where each arrow denotes that the previous class is the proper subclass of the latter one. In this Chapter we deal with the questions of the inheritance of properties (denoted by, e.g, P ) of topological vector spaces, when passing to a subspace or a quotient of the given space, and also with a kind of a converse question, the so- called three-space-problem (abbrev. 3SP) which can be formulated in the following way: Let F be a closed subspace of an tvs (E, t) and let

q (∗) 0 → (F, t|F ) →i (E, t) → (E/F, t/F ) → 0 be the corresponding short exact sequence (i and q are canonical embedding and quotient mapping, respectively). If the spaces (F, t|F ) and (E/F, t/F ) possess a certain property P , does (E, t) have to possess the same property? If the answer 40 II. Subspaces and quotients of topological vector spaces to this question is positive, we say that the property P is three-space stable (3SP- stable). In the present paragraph P will denote the property of belonging to one of the mentioned five basic classes of lcs’s. Then it is trivial to show that each property P is inherited to every quotient by a closed subspace. In the case when P denotes the property of “being a Mackey space”, see Corollary of the Proposition I.6.13. However, in general, none of these properties is inherited by an arbitrary, even closed subspace [62], [103]. We give an example of that kind. Example. As we have shown, there exist normed (non-complete) spaces which are not barrelled. Such a space is a (dense) subspace of its completion, which is barrelled as all Banach spaces are. To construct an example of a closed non- barrrelled subspace of a barrelled space, consider a complete lcs (E, t) which is not quasibarrelled (such spaces exist — we can take, e.g., a non-reflexive (F)-space G and denote (E, t) := (G0, τ(G0,G)), see [62], 27.1). Using the proposition I.4.2, (E, t) is embedded as a closed subspace of a product of Banach spaces. But the last space is barrelled as all products of barrelled spaces are ([62], 27.1). It appeared that the problem of inheritance of properties P to nonclosed subspaces of finite codimension or, equivalently (using mathematical induction), to dense subspaces of the codimension 1 (dense hyperplanes), is nontrivial and one of the important questions in the general theory of locally convex spaces. We shall give chronologically the results concerning the problem of subspaces for the basic properties.

Theorem 1.1. (J. Dieudonn´e[34]) The class of barrelled (resp. bornological) lcs’s is stable with respect to arbitrary ﬁnite-codimensional subspaces.

The stated result is a direct consequence of the following two propositions.

Proposition 1.2. (Y. K¯omura [61]; M. De Wilde [132]) Let (E, t) be an lcs and (F, t|F ) its subspace of finite codimension. Then for each barrel TF in the subspace (F, t|F ) there exists a barrel TE in the space (E, t) such that TE ∩F = TF . Proof. It is sufficient to consider the case when F is a hyperplane, and then use the mathematical induction. So, let F be a hyperplane in E , x0 ∈ E \ F a fixed point, U = { λx0 | |λ| 6 1 } and TF a barrel in the subspace (F, t|F ). Then the wanted barrel TE in (E, t) is obviously given by

( t TF + U, if T F ⊂ F , TE = t t T F , if T F 6⊂ F .

As we saw at the end of the previous Chapter, if (E, t) is an lcs and F its arbitrary subspace, then the topology β(E,E0)|F is in general coarser than the strong topology β(F,F 0) on the subspace. However, if the codimension of the subspace F is ﬁnite, we obtain 2.1. Subspaces of lcs’s belonging to the basic classes 41

Corollary. The strong topology β(E,E0) of an lcs (E, t) induces the strong topology β(F,F 0) on its ﬁnite-codimensional subspace F . Proof. A base of neighbourhoods of the origin for the topology β(F,F 0) is formed by all the barrels of the space (F, t|F ), and so the given assertion is a consequence of the Proposition 1.2.

Proposition 1.3. (M. De Wilde [132]) Let (E, t) be an lcs and (F, t|F ) its subspace of ﬁnite codimension. Then for each t|F -bornivorous absolutely convex subset UF of F there exists a t-bornivorous absolutely convex subset VE of E such that VE ∩ F = UF .

Proof. If F is a subspace of codimension 1 and x0 ∈ E \ F , then each x ∈ E can be uniquely represented as x = y + λx0 , y ∈ F , λ ∈ K. If we put y = P x, we obtain a linear operator from E into F . Now consider the two possible cases: 1◦ The projection P maps t-bounded subsets onto t|F -bounded ones. Then, −1 clearly, VE = P UF is an absolutely convex and t-bornivorous subset of E , such that VE ∩ F = UF . 2◦ The projection P does not map all t-bounded subsets of E onto t|F - bounded ones. Then there exists a t-bounded sequence (xn) in the space E such that (P xn) is not t|F -bounded. Now we obtain that xn = P xn + λnx0 , where it can be supposed that λn → ∞. Let, furthermore, B0 = Γ({x0} ∪ { xn | n ∈ N }). 1 1 Taking yn = − P xn , we get that yn ∈ F and yn − x0 ∈ B0 for all n ∈ N. λn λn Let now B be the collection of all absolutely convex t-bounded subsets of E which contain B0 . In order that a subset of E be t-bornivorous, it is suﬃcient that it absorbs the elements of the family B . Choose for each B ∈ B , cB 6= 0 such that

2cBB ∩ F ⊂ UF . ¡ ¢ 0 S Putting U = Γ B∈B cBB , we obtain an absolutely convex and t-bornivorous 0 0 set in E . We have also that U ∩F ⊂ UF . Since Γ(UF ∪U )∩F = UF , the wanted t-bornivorous subset VE of E is given by

0 VE = Γ(UF ∪ U ).

If the codimension of F is greater than one, the mathematical induction can be used to complete the proof. To each lcs (E, t) a Mackey space (E, τ(E, E˜0)) can be corresponded, where E˜0 is the vector space of all linear functionals on E which are bounded on t-bounded subsets. A base of neighbourhoods of the origin for the topology τ(E, E˜0) is formed by all absolutely convex and t-bornivorous subsets of E (see also the proof of the Proposition I.5.10). The space (E, τ(E, E˜0)) (which is clearly bornological) is called “the bornological space associated to the given lcs (E, t)”. By the Proposition 1.3, it follows 42 II. Subspaces and quotients of topological vector spaces

Corollary. The associated bornological topology τ(E, E˜0) of an lcs (E, t) induces on F the bornological topology τ(F, F˜0) associated to its finite-codimensional subspace (F, t|F ). Theorem 1.4. (M. Valdivia [112]) The class of quasibarrelled lcs’s is stable with respect to arbitrary finite-codimensional subspaces. This theorem is a direct consequence of the following Proposition 1.5. (M. Valdivia [112]) Let (E, t) be an lcs and (F, t|F ) its subspace of finite codimension. Then for each t|F -bornivorous barrel TF of the subspace F there exists a t-bornivorous barrel TE of the space (E, t), such that TE ∩ F = TF .

Proof. It is suﬃcient to prove that the barrel TE obtained in the proof of the Proposition 1.2 is t-bornivorous, if the barrel TF is t|F -bornivorous. Using the Corollary of the Proposition 1.3, in the case when the subspace (F, τ(F, F˜0)) τ(E,E˜0) is dense in the space (E, τ(E, E˜0)), we obtain that T is a τ(E, E˜0)- τ(E,E˜0) t neighbourhood of the origin and then, since T ⊂ T F , it follows that TE is a t-bornivorous barrel in (E, t). If (F, τ(F, F˜0)) is a closed subspace of (E, τ(E, E˜0)), 0 then TF + U is also a τ(E, E˜ )-neighbourhood of the origin , and hence a t- bornivorous barrel of the space (E, t) in the case that (F, t|F ) is a closed subspace of (E, t). Finally, if (F, t|F ) is a dense subspace of (E, t), then

( t t TF + U, if T F ⊂ F , TE = T F + U = t t T F , if T F 6⊂ F . is again a t-bornivorius barrel in (E, t). Corollary. The topology β∗(E,E0) of an lcs (E, t) induces the topology β∗(F,F 0) on each finite-codimensional subspace (F, t|F ) of the space (E, t). Proof. A base of neighbourhoods of the origin of the topology β∗(F,F 0) is formed by all t|F -bornivorous barrels and so the proof follows by the Proposition. Contrary to barrelled, bornological and quasibarrelled spaces, the classes of ultrabornologique and Mackey spaces are not stable for passing to a dense hyperplane. For ultrabornologique spaces even more has been proved. Theorem 1.6. (M. Valdivia [121]) If (E, t) is an ultrabornologique lcs whose topology t is distinct from the finest locally convex topology on E , then there exists a t-dense hyperplane F in E such that the subspace (F, t|F ) is not ultrabornologique. The proof of the theorem can be found in [121]. Observe only that from that proof one can also conclude that in each ultrabornologique space (E, t), where t 6= τ(E,E∗) (the finest locally convex topology), there exists a dense hyperplane F all of whose Banach discs (i.e. absolutely convex sets B such that (EB, pB), are Banach spaces, see the proof of the Proposition I.5.10) are of finite dimension. 2.1. Subspaces of lcs’s belonging to the basic classes 43

In connection with bornological and ultrabornologique spaces the following properties are of interest. Proposition 1.7. Let U be an absolutely convex subset of an lcs (E, t). Then the following conditions are equivalent: 1◦ U is a t-bornivorous set; 2◦ U absorbs all t-precompact subsets; 3◦ U absorbs all t-compact subsets; 4◦ U absorbs all t-zero sequences. Proposition 1.8. Let U be an absolutely convex subset of an lcs (E, t). Then the following conditions are equivalent: 1◦ U absorbs all t-Banach discs; 2◦ U absorbs all σ(E,E0)-compact absolutely convex subsets; 3◦ U absorbs all t-compact absolutely convex subsets. Proof. The implications 1◦ =⇒ 2◦ =⇒ 3◦ are obvious. Let us prove that 3◦ =⇒ 1◦ . If U does not absorb a certain t-Banach disc A, then A 6⊂ n2U for 1 each n ∈ N, i.e. if we choose yn = n an , for arbitrary an ∈ A, then yn ∈/ nU , n ∈ N. But the sequence (yn) converges to the origin in the norm of the space EA . Furthermore, EA Γ({ yn | n ∈ N }) ⊂ A ⊂ EA,

EA wherefrom it follows that Γ({ yn | n ∈ N }) is a compact and absolutely convex subset of the Banach space EA and, a fortiori, a t-compact subset of E , which is not absorbed by U . A contradiction. The following example shows that the class of Mackey spaces is not stable for passing to a dense hyperplane. Example. (J. Shmets [104]) Let E be a Banach space of inﬁnite dimension and x00 a linear functional on E0 which is not β(E0,E)-continuous. Then ker x00 = { x0 ∈ E0 | x00(x0) = 0 } is a β(E0,E)-dense hyperplane in the space E0 . So, there 0 0 0 00 0 exists x0 ∈ E such that E = ker x + span x0 (algebraic direct sum). Then the 00 0 spaces E and ker x forms a dual pair and the hyperplane ker x0 is dense with respect to the Mackey topology τ(E, ker x00) and the space

0 00 0 (ker x0, τ(E, ker x )| ker x0) is not a Mackey space. In the sequel we state the results concerning the inheritance of the basic properties to subspaces whose codimension is not ﬁnite. Theorem 1.9. (M. Valdivia [113]; S. Saxon-M. Levin [99]) The class of barrelled lcs’s is stable with respect to each subspace of countable inﬁnite codimension. The theorem is a direct consequence of the following 44 II. Subspaces and quotients of topological vector spaces

Proposition 1.10. (J. H. Webb [131]) Let (E, t) be a barrelled lcs and (F, t|F ) its subspace of countable infinite codimension. Then for each t|F -barrel TF of the subspace (F, t|F ) there exists an absolutely convex absorbing subset TE t of the space E such that T E ⊂ 2TE and TE ∩ F = TF . Proof. Since the subspace F is of countable infinite codimension in the vector space E , there exists a sequence (En) of subspaces of E such that En is of S∞ codimension 1 in En+1 , n ∈ N, E1 = F and E = n=1 En . By the proposition 1.2 there exists a sequence (Tn) of barrels in En (T1 = TF ) such that Tn+1 ∩En = Tn , S∞ n ∈ N. Let TE = n=1 Tn . TE is clearly an absorbing and absolutely convex t subset of E . Since TE ∩ F = TF , we have only to show that T E ⊂ 2TE . Let z∈ / 2TE . Then z ∈ Em for m great enough. For such m, z∈ / 2Tm 0 and by a corollary of the Hahn-Banach theorem there exists fm ∈ Em such that fm(z) = 2 and |fm(x)| 6 1 for all x ∈ Tm . If each linear functional fm is extended 0 to E , then we obtain a sequence (fm) which is σ(E ,E)-bounded. Since the space 0 (E, t) is barrelled, then (fm) is a t-equicontinuous subset, i.e. there exists f0 ∈ E such that f0(z) = 2 and |f0(x)| 6 1 for all x ∈ TE (f0 is a weak cluster point). Hence, z∈ / T E , in other words T E ⊂ 2TE . Some results for subspaces of (possibly) uncountable codimension were obtained, too. Proposition 1.11. (M. Valdivia, see [79], Nota 1) If (E, t) is a barrelled lcs, such that the space (E0, τ(E0,E)) is complete, then each subspace F of E with the codimension less than 2ℵ0 is barrelled in the induced topology. Examples. (S. Saxon-M. Levin [99]) 1◦ Let s be the space of all real sequences with the product topology. The subspace m consisting of all bounded sequences is dense in s and it is not barrelled since the barrel { (xn)n∈N | kxnk 6 1 } is not a neighbourhood of the origin in the induced topology. The codimension of m in s is uncountable. 2◦ The linear span L of a basis in a Banach space E is of countable dimension and it is not a barrelled space. L is obviously of uncountable codimension in E and so there must exist an increasing sequence (Ln) of subspaces of E such thatS Ln is of uncountable codimension in Ln+1 for each n ∈ N, L1 = L and E = n∈N Ln . Since E is a Baire space, E cannot be represented as a countable union of nowhere dense sets. Thus, some Ln is a Baire, i.e. a barrelled space. It follows from 1◦ and 2◦ that some of the dense subspaces of uncountable codimension of barrelled spaces are barrelled and some are not. The following example shows that the classes of bornological and quasibarrelled spaces are not stable with respect to subspaces of countable infinite codimension. Example. (M. Valdivia [117]) From [41] we know that there exists an lcs (E, t) which is a strict inductive limit of an increasing sequence (En) of separable Fréchet spaces, such that there exists a non-closed subspace G in E such that G ∩ En is closed in En , n ∈ N. Let An be a countable dense set in En , P the linear space 2.1. Subspaces of lcs’s belonging to the basic classes 45

S∞ generated by n=1 An and F the linear span of P ∪ G. Since P is a sequentially dense subspace of (E, t), it is bornological by [116]. Subspace G is not barrelled ([117], Theorem f), p. 22) and since it is quasicomplete, it is not quasibarrelled either (Corollary of the Proposition I.6.8). It follows from the fact that P has a countable basis, that G is a subspace of P of countable infinite codimension. Thus, F is a bornological space which has the subspace G of countable infinite codimension, such that G is not quasibarrelled. Consider now an lcs (E, t) and its subspace (F, t|F ) of countable infinite codimension which satisfies the condition: (B) For each t-bounded subset B of E , F is of finite codimension in the space span(F ∪ B). Theorem 1.12. (J. H. Webb [129]) The class of bornological (resp. quasibarrelled) lcs’s with the property (B) is stable with respect to subspaces of countable infinite codimension.

Proof. Let U0 be a t|F -bornivorous absolutely convex set in F and (en) a sequence in E (linearly independent modulo F ) such that E = F ⊕ span{ en | n ∈ N }. Let F0 = F , Fn = span(Fn−1 ∪ {en}), n ∈ N. By the Proposition 1.3 there exist absolutely convex sets Ui ⊂ Fi which are t|Fi -bornivorous and S∞ Ui ∩ Fi−1 = Ui−1 , i = 1, 2,... . Putting V = i=1 Ui , we get an absolutely convex t-bornivorous set in E (by the condition (B) each t-bornivorous subset is contained in some Fi ) such that V ∩ Fi = Ui , i = 1, 2,... and V ∩ F = U0 . It follows that U0 is a t|F -neighbourhood of the origin, i.e. (F, t|F ) is a bornological space since V is a t-neighbourhood of the origin. If (E, t) is a quasibarrelled space with the property (B), by the Proposition S∞ 1.5, we obtain the same set V = i=1 Ui , where Ui is a bornivorous barrel in the space (Fi, t|Fi). That V ⊂ 2V can be proved as in the Proposition 1.10. Interesting results about quotients and countable-codimensional subspaces of Fréchet and Banach spaces were obtained recently by S. A. Saxon ([97], [98]), some of them answering questions posed in [81]. We shall give a brief account of these results. We start with Saxon’s proofs of two classical results. The space which is a product of countably many copies of the scalar field will be denoted by ω . Theorem 1.13. (M. Eidelheit [37]) Every non-normable Fréchetlcs (E, t) has a quotient isomorphic to ω .

Proof. Let (Un) be a basic closed topological string in (E, t). Since U1 0 0 is not (σ(E,E )-) bounded, there exists f1 ∈ E which is unbounded on U1 . 0 Continuing by induction, if f1 , ... , fn ∈ E have been chosen so that fi is T ⊥ T ⊥ unbounded on Ui ∩ ( j

T on U ∩( f ⊥). Let now (λ ) be an arbitrary sequence of scalars. Again using n j1 xj converges to some x in Fr´echet space E and fn(x) = fn(x1 + ··· + xn) = λn for n = 1, 2,... , hence ∞ the mapping y 7→ (fn(y))n=1 , y ∈ E is linear, continuous and onto ω ; applying the open mapping theorem, we obtain the desired result.

Theorem 1.14. (S. Banach, S. Mazur [7]) Each separable Banach space F is a quotient of the sequence space l1 . 1 Proof. Let e1 , e2 , ... be unit vectors in l ; obviously the linear span of 1 {e1, e2,... } is dense in l . Let B be the unit ball in F and B0 = {h1, h2,... } a 1 dense subset of B . Deﬁne the continuous linear mapping T : l → F by T (ei) = hi , i = 1, 2,... . Then T maps the unit ball of l1 onto a dense subset of B . By the open mapping theorem T is open and so F is isomorphic to a certain quotient of l1 . An lcs E is called regenerative if all of its countable-codimensional subspaces have quotients isomorphic to E . For the proof of the next two results see [97] and [98].

p Proposition 1.15. The sequence spaces ω , c0 and l , 1 6 p 6 ∞ are regenerative.

Proposition 1.16. Let E and F be Fr´echetlcs’s such that, for some subspace M of E , E/M is isomorphic to F , and let E0 be a countable-codimensional subspace of E . Then there exists a subspace F0 of F with the codimension not exceeding that of E0 such that some quotient of E0 is isomorphic with F0 .

Proposition 1.17. [98] Let E and F be Fr´echetspaces such that F is regenerative and some quotient of E is isomorphic to F . If E0 is a countable- codimensional subspace of E , then some quotient of E0 is also isomorphic to F .

Proof. By the Proposition 1.16, a quotient of E0 is isomorphic to a countable- codimensional subspace F0 of F . Since F0 has a quotient isomorphic to F , and since quotients are “transitive” (i.e. a quotient of a quotient is again a quotient), E0 has a quotient isomorhic to F .

Theorem 1.18. (S. A. Saxon [97]) Every countable-codimensonal subspace of a non-normable Fr´echetlcs has a quotient isomorphic to ω . Proof follows by combining the Theorem 1.13, regenerativity of ω and the Proposition 1.17.

Theorem 1.19. (S. A. Saxon [98]) Each separable Banach space E is isomorphic to some quotient of any countable-codimensional subspace of l1 . Proof follows again by using the transitivity of quotients, the regenerativity of l1 and the Theorem 1.14. 2.2. Subspaces of lcs’s from some other classes 47

We shall ﬁnish this paragraph by stating some questions and results concerning subspaces of Banach and Baire lcs’s. As is well known, an lcs (E, t) is Baire if E is not a union of an (increasing) sequence of nowhere dense subsets. The following problem was open for a long time: is each dense hyperplane of a Baire lcs always a Baire space? In 1966 V. L. Klee and A. Wilansky [60] posed the following question: does a hyperplane in a Banach space have to be either closed or of second category? This question is equivalent to the following: is a dense hyperplane in a Banach space always a Baire space? A. R. Todd and S. Saxon stated in [106] the following four equivalent assertions under the name of Wilansky-Klee Conjecture: (i) A linear form, diﬀerent from zero, on a Banach space is continuous if and only if its kernel can be represented as a countable union of nowhere dense sets. (ii) Each dense hyperplane of a Banach space is a Baire space. (iii) Each dense hyperplane in a locally convex Baire space is a Baire space. (iv) Each countably-codimensional subspace of a locally convex Baire space is a Baire space. The answer to the Wilansky-Klee Conjecture is negative, by the following results.

Theorem 1.20. (J. Arias de Reyna [6]) If Martin’s axiom [105] holds, then in each separable inﬁnite-dimensional Banach space there exists a dense hyperplane of the ﬁrst category.

Theorem 1.21. (M. Valdivia [122]) Each separable inﬁnite-dimensional Baire space has a dense hyperplane of the ﬁrst category.

At the end we state a question concerning barrelledness of a subspace: is there a Banach space of infinite dimension such that all of its dense subspaces are barrelled? This question is in connection with the following result of Saxon- Wilansky [102]: Let E be an infinite-dimensional Banach space. Then E has an infinite-dimensional separable quotient if and only if E has a dense subspace which is not barrelled.

2.2. SUBSPACES OF LCS ’S FROM SOME OTHER CLASSES

There are a lot of generalizations of the barrelledness concept which proved themselves to be useful in the theory of locally convex spaces. We shall introduce now some of them. It can be easily shown that all of these new notions determine the classes of lcs’s which are all different from each other and from classes introduced in the previous paragraph. Let (E, t) be an lcs. By a σ -barrel (resp. d-barrel) in (E, t) we mean a T∞ barrel V = n=1 Vn where Vn are t-closed absolutely convex t-neighbourhoods of the origin (resp. σ(E,E0)-closed absolutely convex σ(E,E0)-neighbourhoods 48 II. Subspaces and quotients of topological vector spaces of the origin). The space (E, t) is called countably barrelled (resp. ω -barrelled) if each σ -barrel (resp. d-barrel) in it is a t-neighbourhood of the origin . For both classes we can give dual characterizations: (E, t) is countably barrelled (resp. ω -barrelled) if and only if each σ(E0,E)-bounded set which is a countable union of t-equicontinuous subsets (resp. each σ(E0,E)-bounded sequence) is t- equicontinuous. An lcs (E, t) is sequentially barrelled (c0 -barrelled in some papers) if each σ(E0,E)-zero sequence is t-equicontinuous. Similarly countably quasibarrelled and ω -quasibarrelled spaces are defined. These are the spaces in which each bornivorous σ -barrel (resp. bornivorous d- barrel) is a t-neighbourhood of the origin. A space (E, t) is sequentially quasibarrelled if each β(E0,E)-zero sequence is t-equicontinuous. Dual characterizations of countably and ω -quasibarrelled spaces are similar as for countably and ω -barrelled ones, only instead of σ(E0,E), the strong topology β(E0,E) in the dual space E0 is used. The space (E, t) is sequentially quasibarrelled if and only if for every 0 0 sequence (Vn) of σ(E,E )-closed absolutely convex σ(E,E )-neighbourhoods of T∞ the origin inE such that every bounded subset of E is contained is n=m Vn for T∞ some m, the set n=1 Vn is a t-neighbourhood of the origin [127]. An lcs (E, t) is of the type (DF) if it is countably quasibarrelled and has a fundamental sequence of t-bounded subsets. Such are, e.g., the strong duals of Fréchet lcs’s. All these types of spaces are kinds of generalizations of barrelled spaces. For other types of generalizations of such spaces, we shall introduce some notions. A barrel T in an lcs (E, t) is called a b-barrel (resp. k -barrel, p-barrel) if it intersects each bounded and closed (resp. compact, precompact) absolutely convex subset by a t-neighbourhood of the origin in the induced topology. Then the space (E, t) is called b-barrelled (resp. k -barrelled, p-barrelled) if each b-barrel (resp. k -barrel, p-barrel) in it is a t-neighbourhood of the origin. If we replace barrels by absolutely convex sets, we obtain b-spaces (resp. k -spaces (or Kelley spaces), 0 p-spaces). An lcs (E, t) is a b -space (resp. kr -space, pr -space) if each linear form on it is continuous, i.e. belongs to the dual E0 , whenever its restrictions to all t-bounded and closed (resp. compact, precompact) absolutely convex subsets are continuous. Finally, the space (E, t) is said to be of the type (Db ) (in the sense of Noureddine) if it is b-barrelled and has a fundamental sequence of t-bounded subsets. All mentioned classes of spaces are stable with respect to arbitrary quotients by closed subspaces, and none is stable for passing to arbitrary, even closed subspaces. When subspaces of finite codimension are concerned, first of all we state the following result.

Theorem 2.1. [87] Let F be a hyperplane of an lcs (E, t). For each σ -barrel (resp. bornivorous σ -barrel, d-barrel, bornivorous d-barrel) TF in the subspace (F, t|F ) there exists a σ -barrel (resp. bornivorous σ -barrel, d-barrel, bornivorous d-barrel) TE in the space (E, t) such that TE ∩ F = TF . T∞ 0 Proof. If TF = n=1 Vn is a σ -barrel in (F, t|F ), then TF is a β(F,F )- neighbourhood of the origin. There are two possible cases: 2.2. Subspaces of lcs’s from some other classes 49

1◦ (F, β(F,F 0)) is a dense subspace of (E, β(E,E0)). Then (F, t|F ) is dense T t t β(E,E0) T t in (E, t). Since n∈N V n ⊃ V ⊃ V , TE = n∈N V n is a σ -barrel in (E, t) such that TE ∩ F = TF . ◦ 0 0 0 2 (F, β(F,F )) is closed in (E, β(E,E )). Then TF + C is a β(E,E )- neighbourhood of the origin, where C is the circled cover of theT set {x}, x ∈ E \F . If (F, t|F ) is a closed subspace of (E, t), then obviously TE = n∈N(Vn + C) is a σ -barel in (E, t) whose intersection with F is TF . In the case when (F, t|F ) is a dense subspace of (E, t), we distinguish two more cases:

\ t \ t (a) V n 6⊂ F and (b) V n ⊂ F. n∈N n∈N

T t T t In the case (a) we can take TE = n∈N V n and in the case (b) TE = n∈N(V n + C). In the latter case for proving that TE ∩ F = TF we can use the fact that sequence (Vn) can be considered decreasing (see [130]). This ﬁnishes the proof for σ -barrells. T If TF = n∈N Vn is a d-barrel in (F, t|F ), then TF is a σ -barrel in the space (F, σ(F,F 0)). Since the weak topology σ(E,E0) induces on F the weak topology σ(F,F 0), the proof is nearly the same as for the σ -barrel, only instead of topologies t and t|F , the topologies σ(E,E0) and σ(F,F 0) = σ(E,E0)|F are used. For a bornivorous σ -barrel instead of the topologies β(E,E0) and β(F,F 0), the topologies τ(E, E˜0) or β∗(E,E0) and τ(F, F˜0) or β∗(F,F 0) are used (they have the same bounded subsets). Since a bornivorous d-barrel of the space (E, t) is a bornivorous σ -barrel of the space (E, σ(E,E0)), the proof for it is the same as for the bornivorous σ -barrel. So the proof of the theorem is complete.

Corollary. The class of countably barrelled (resp. ω -barrelled, countably quasibarrelled, ω -quasibarrelled, (DF)-) spaces is stable with respect to subspaces of ﬁnite codimension.

Example. (J. H. Webb [127]) Let (E, t) be a sequentially barrelled space such that the space (E0, σ(E0,E)) is not sequentially complete. An example of 1 1 0 such a space can be (l , τ(l , c0)). Let (fn) be a σ(E ,E)-Cauchy sequence which 0 0 ∗ 0 is not σ(E ,E)-convergent in E . Since fn →σ(E∗,E) f ∈ E \ E , F = { x ∈ E | 0 0 f(x) = 0 } is a dense hyperplane in E . Thus, (F, t|F ) = E . The sequence (fn) is clearly a σ(E0,F )-zero sequence, but it is not t|F -equicontinuous, which means that (F, t|F ) is not a sequentially barrelled space. This shows that the class of sequentially barrelled spaces is not stable even with respect to dense hyperplanes. For sequentially quasibarrelled spaces we give the following result.

Theorem 2.2. [87] The class of sequentially quasibarrelled spaces is stable for passing to subspaces of ﬁnite codimension. Before proving this theorem, we shall state (for the proof see [116]) a lemma which will appear to be useful in many considerations concerning subspaces of lcs’s. 50 II. Subspaces and quotients of topological vector spaces

Lemma 2.3. (M. Valdivia [116]) Let (E, t) be an lcs and (F, t|F ) its subspace of codimension 1. Let M be a family of subsets of E satisfying the following conditions: 1◦ if M ∈ M, then M is bounded, closed and absolutely convex; ◦ 2 if M1,M2 ∈ M, then there is M3 ∈ M such that M1 ∪ M2 ⊂ M3 ; 3◦ if M ∈ M, then λM ∈ M for all λ ∈ K, λ 6= 0. If there exists M ∈ M such that M ∩ F is not closed, then for each P ∈ M there t exists Q ∈ M, such that P ⊂ Q ∩ F . Remarks. If the intersection M ∩F is a closed subset of E for each M ∈ M, then it is said that the hyperplane F is almost closed in E with respect to the family M; in the oposite case, i.e. when for each P ∈ M there exists Q ∈ M, such t that P ⊂ Q ∩ F , the hyperplane F is called ultra-dense with respect to M. In particular, when M is equal to the family B(E) of all closed bounded absolutely convex subsets of E (or to its fundamental subfamily), the corresponding terms will be just almost closed and ultra-dense. It can be proved that each dense hyperplane F of an lcs (E, t) is ultra-dense 0 with respect to the family M if the space (E , tM) is complete. Moreover, in this 0 0 0 case (E , tM) = (F , tM1 ) = (E , tM1 ) holds, where M1 = { M ∩ F | M ∈ M }. Taking M to be the family of all bounded (resp. strongly bounded, precompact) absolutely convex and closed subsets of (E, t), we obtain that the hyperplane (F, t|F ) is either almost closed or ultra-dense with respect to each of these families. Proof of the Theorem 2.2. It suffices to prove the theorem for the case of a dense hyperplane. So, let (E, t) be a sequentially quasibarrelled space and let (F, t|F ) be its dense hyperplane. By the preceding Lemma, we know that F is either ultra-dense or almost closed in (E, t). In the first case we have β(E0,E) = β(F 0,E) = β(F 0,F ) and so (F, t|F ) is obviously sequentially quasibarrelled if and only if (E, t) is such. If F is almost closed, then F = f −1(0), where f is a sequentially continuous linear form on E ; it defines two sequentially continuous, hence bounded, projections:

pi : E → Fi, i = 1, 2 (F1 = F,F2 = L — the complementary line). 0 0 Let (Vn) be a sequence of σ(F,F )-closed absolutely convex σ(F,F )-neighbourhoods of the origin, such that for each t|F -bounded set A there exists m ∈ N so T∞ T∞ that A ⊂ n=m Vn . We have to prove that n=1 Vn is a t|F -neighbourhood of the origin. Let now B be an arbitrary t-bounded subset of E ; then for some k1 ∈ N and λ > 0: ∞ ∞ µ ∞ ¶ \ \ t \ t B ⊂ p1(B) + p2(B) ⊂ Vn + λC ⊂ V n + λC ⊂ max{1, λ} V n + C ,

n=k1 n=k1 n=k1 where C = { µx | |µ| 6 1 }, x ∈ L. Now we consider two cases: \∞ \∞ ◦ t ◦ t 1 (∃m0 ∈ N) V n 6⊂ F and 2 V n ⊂ F for each m ∈ N. n=m0 n=m 2.2. Subspaces of lcs’s from some other classes 51

T t In the ﬁrst case, ∞ V is an absorbing set in E and so there exists µ > 0 n=m0 n T∞ t t such that B ⊂ µ n=k V n , where k = max{m0, k1}. Since (µV n)n∈N is a sequence of σ(E,E0)-closed absolutely convex σ(E,E0)-neighbourhoods of the origin , using the dual characterization of sequentially quasibarrelled spaces, we T∞ t T∞ t obtain that n=1(µV n), and so also n=1 V n , are t-neighbourhoods of the origin. ¡T∞ t ¢ T∞ So, F ∩ n=1 V n = n=1 Vn is a t|F -neighbourhood of the origin. In the second case we have:

∞ ∞ \ t \ t B ⊂ max{1, λ} (V n + C) = µ(V n + C),

n=k1 n=k1

t 0 where µ = max{1, λ}. Now, (µ(V n + C))n∈N is obviously a sequence of σ(E,E )- closed absolutely convex σ(E,E0)-neighbourhoods of the origin and using the same T∞ t argument as in the ﬁrst case we obtain that n=1(V n + C) is a t-neighbourhood ¡T∞ t ¢ T∞ of the origin. The proof that n=1(V n + C) ∩ F = n=1 Vn is the same as in the Theorem 2.1. The theorem is proved. Remark. Combining the previous theorem and example we conclude that there exists a sequentially quasibarrelled space which is not sequentially barrelled. For a countably barrelled or ω -barrelled lcs (E, t), the weak dual (E0, σ(E0,E)) is sequentially complete and it is said that such spaces have the property (S). However, sequentially barrelled spaces are not always with the property (S). For spaces of mentioned classes we give the following results.

Theorem 2.4. (M. Levin-S. Saxon [67]; M. Valdivia [120]; J. H. Webb [128]) The class of countably barrelled (resp. ω -barrelled) spaces is stable with the respect to subspaces of countable inﬁnite codimension.

The proof is based on the following

Lemma 2.5. (S. Saxon-M. Levin [99]; J. H. Webb [128]) Let (E, t) be an lcs with the property (S). If A is a closed absolutely convex subset of E , such that the codimension in E of span(A) is countable inﬁnite, then span(A) is t-closed.

Proof. Let E = span(A) ⊕ span{ xn | n ∈ N }, where (xn) is a linearly 0 independent sequence. Let us construct a sequence of gk ∈ E such that gk(xi) = δki and gk(a) = 0 for each a ∈ A. The construction is as follows: let

Br = Γ(A ∪ {x1, . . . , xk−1, xk+1, . . . , xr}), r > k.

Then Br is absolutely convex and closed and xk ∈/ rBr . By the Hahn-Banach 0 1 theorem there exist fr such that fr(xk) = 1 and |fr(x)| 6 r for all x ∈ Br . 0 The sequence (fr)r>k is clearly a σ(E ,E)-Cauchy sequence, which means that it 0 converges to some gk ∈ E . So we have constructed the sequence (gk). As far as T∞ −1 span A = k=1 gk (0), we conclude that span A is closed. 52 II. Subspaces and quotients of topological vector spaces

We state also the following result on sequentially barrelled spaces.

Theorem 2.6. (J. M. Garcia-Lafuente [39]) The class of sequentially barrelled spaces with the property (S) is stable with respect to subspaces of countable codimension. In what follows we give results about subspaces of b-barrelled (resp. p- barrelled, k -barrelled) spaces, b-spaces (resp. p-spaces, k -spaces) and Db -spaces. All these spaces have dual characterizations, too. E.g., an lcs (E, t) is b-barrelled if and only if each subset A ⊂ E0 is t-equicontinuous if its restriction to each t- bounded absolutely convex subset is t-equicontinuous. For p-barrelled spaces even 0 more holds: an lcs (E, t) is p-barrelled if and only if each Ep -precompact subset is t-equicontinuous. However, for b-barrelled (resp. k -barrelled) spaces we only have 0 0 that each β(E ,E)-precompact (resp. Ec -precompact) subset is t-equicontinuous. Further, an lcs (E, t) is a b-space (resp. p-space, k -space) if and only if (E, t) 0 is b-barrelled and a b -space (resp. p-barrelled and a pr -space; k -barrelled and a kr -space). That is why a space of the type (DF) is of the type Db , too. Using the given characterizations, the following results can be proved.

Proposition 2.7. The class of p-spaces is stable with respect to subspaces of ﬁnite codimension. Proof. Since this class is stable with respect to quotients by closed subspaces, it suﬃces to prove the proposition in the case of a dense hyperplane. Let F be a dense hyperplane in a p-space (E, t). By [116], Theorem 2, the subspace (F, t|F ) is a pr - space and so it is enough to prove that (F, t|F ) is p-barrelled. Let T ⊂ F 0 = E0 be such a subset that its restriction TP to each precompact absolutely convex subset P of the subspace (F, t|F ) is t|F -equicontinuous. This means that there exists a t|F -neighbourhood of the origin U such that

◦ ◦ T ⊂ (P ∩ U)◦ ⊂ P ◦ + U ◦ = P + U ⊂ 2(U ∩ P )◦ for each precompact subset of (F, t|F ). The last inclusion implies that the restric- 1 tion of the set 2 T to each t-precompact subset is t-equicontinuous, wherefrom 1 the t-equicontinuity of the set 2 T , and so also of T , follows. Thus, (F, t|F ) is a p-barrelled space since E0 and F 0 = E0 have the same equicontinuous subsets. Remark. In the same way the stability of the class of b-spaces with respect to finite-codimensional subspaces can be proved. Of course, the same is true for b-barrelled and p-barrelled spaces. A more general result can also be proved: each b-barrel (resp. p-barrel) of a finite-codimensional subspace has its extension to a barrel of the same type in the whole space. Thence it follows that a finite- codimensional subspace of a Db -space is also a Db -space. When k -spaces (k -barreled spaces) are concerned, similarly as for ultrabornologique spaces, the following negative result can be proved. 2.2. Subspaces of lcs’s from some other classes 53

Theorem 2.8. [82] Let (E, t) be an lcs which satisﬁes the following conditions: 1◦ (E, t) is k -barrelled; 2◦ (E, t) is not quasibarrelled; 3◦ (E, t)0 = (E, τ(E, E˜0))0 , where (E, τ(E, E˜0)) is the ultrabornologique space associated to (E, t). Then there exists a dense hyperplane F ⊂ E such that (F, t|F ) is not a k - barrelled space. For the proof, the following statements are used. Their proofs are easy.

Proposition 2.9. Let (E, t) be an lcs. Then (E, τ(E, E˜0)) is the ﬁnest locally convex topology on E if and only if dim EK < ∞ for each t-compact absolutely convex subset K .

Proposition 2.10. If dim EK < ∞ for each t-compact absolutely convex subset K , then (E, t) is barrelled if and only if it is k -barrelled.

Proposition 2.11. Let (E, t) be a k -barrelled lcs. Then (E, t) is barrelled if and only if it is quasibarrelled.

Proof of the Theorem 2.8. On the base of the given propositions and the conditions of the theorem, we conclude that the topology τ(E, E˜0) is distinct from the finest locally convex topology. Hence, using the remark given after the Theorem 1.6, we obtain that there exists a τ(E, E˜0)-dense hyperplane F of the space (E, τ(E, E˜0)) such that the space (F, τ(E, E˜0)|F ) is not ultrabornologique and that all τ(E, E˜0)|F -compact absolutely convex subsets are of finite dimension. Since the spaces (F, t|F ) and (F, τ(E, E˜0)|F ) have the same compact absolutely convex subsets, it follows (Proposition 2.10) that the space (F, t|F ) is barrelled if and only if it is k -barrelled. But in this case (E, t) would be a barrelled, a fortiori, quasibarrelled space. A contradiction. Example. If (E, k k) is a reflexive Banach space of infinite dimension, then 0 ◦ ◦ ◦ the space Ec satisfies the conditions 1 , 2 and 3 of the previous theorem, where 0 0 Ec is the space E endowed with the topology of uniform convergence on the family of compact absolutely convex subsets of E .

Proposition 2.12. The class of k -spaces is not stable with respect to subspaces of finite codimension. Proof. On the base of the Theorem 1.6 about ultrabornologique spaces and the remark following it, it follows that each Banach space of infinite dimension contains a dense hyperplane which is not ultrabornologique and all of whose compact absolutely convex subsets are of finite dimension. If F is such a hyperplane, then 0 0 Fσ = Fc . In the case when F is a k -space in the norm topology, because of the 0 0 0 ∗ completeness of the space Ec , we have Fσ = Fc = Fσ , which means that F is of finite dimension. A contradiction. 54 II. Subspaces and quotients of topological vector spaces

We shall now use the Lemma 2.3 and the remarks following it to derive some more inheritance properties of lcs’s.

0 Proposition 2.13. The class of b -spaces (resp. pr -spaces) is stable with respect to subspaces of ﬁnite codimension. 0 Proof. An lcs (E, t) is a b -space (resp. pr -space) if and only if its strong 0 0 0 dual Eβ (resp. Ep ) is complete. If now E is a b -space and (F, t|F ) is a dense hyperplane, it is ultra-dense (with respect to the family of all bounded subsets) and 0 0 0 Fβ = Eβ , i.e. (F, t|F ) is a b -space. The case of pr -spaces is treated in the same 0 0 way since then Fp = Ep holds. As far as both classes of spaces are stable with respect to quotients by closed subspaces, the assertion is true for closed hyperplanes, too.

Remark. Each dense hyperplane of a Db -space is ultradense with respect to the family of all bounded subsets. Now we give some hereditary properties of semi-reflexive spaces. This property is not stable for passing to quotients by closed subspaces. Example. G. Kötheand A. Grothendieck constructed an example (see [62] or [103]) of a Fréchet-Montel* space having a closed subspace F such that the quotient E/F is isomorphic to l1 ; in other words the sequence

j q 0 → F → E → (E/F =∼ l1) → 0 is exact, where j and q are canonical embedding and quotient map, respectively. Then E/F is not semireflexive, neither b-reflexive (see the definition below), while E possesses both properties. Furthermore, its strong dual is an example of an ultrabornologique (and so barrelled, bornological and quasibarrelled) space, while its subspace (F ◦, β(E0,E)|F ◦) has neither of these properties. Here also q is an example of a quotient mapping which does not lift bounded sets with closure. On the other hand semi-reflexivity is inherited by all closed subspaces, which follows easily from the Proposition I.6.9. For dense hyperplanes one gets Proposition 2.14. Let (E, t) be a semi-reflexive lcs and (F, t|F ) its dense hyperplane. Then (F, t|F ) is semi-reflexive if it is almost closed and it is not semi- reflexive if it is ultra-dense (with respect to the family of all t-bounded subsets). Proof. In the first case each bounded, closed and absolutely convex subset of the subspace (F, t|F ) is t-closed in E and from the semi-reflexivity of (E, t) it follows that it is also σ(E,E0)-compact, i.e. σ(F,F 0)-compact, which means that (F, t|F ) is semi-reflexive (β(F 0,F ) = τ(F 0,F )). In the second case, using the remark after Lemma 2.3, we get β(E0,E) = β(F 0,E) = β(F 0,F ), wherefrom it follows that (F 0, β(F 0,F ))0 = (E0, β(E0,E))0 = E 6= F , which means that the subspace (F, t|F ) is not semi-reflexive.

*An (ultra)barrelled lcs (tvs) is called a Montel space if every bounded subset of E is relatively compact. It is clear that each Montel lcs is reﬂexive. 2.2. Subspaces of lcs’s from some other classes 55

Corollary. The class of semi-reflexive b0 -spaces is not stable with respect to dense hyperplanes. Proof. All dense hyperplanes of b0 -spaces are ultra-dense. If (E, t) is an lcs, then in the topological dual E0 there exists the unique locally convex topology, denoted by TE0 , such that the space (E0,TE0) is ultrabornologique. A base of neighbourhoods of the origin of this topology is formed by the all so called equivorous subsets of E0 , i.e. those subsets which absorb all t-equicontinuous sets. The topology TE0 is obviously not coarser than the strong topology β(E0,E). So it is natural to consider the class of spaces [10], [26] which satisfy (E0,TE0)0 = E (algebraically) — these spaces are called b-refelexive (inductively semi-reflexive by some authors). Similarly to semi-reflexive spaces (see the previous example), the class of b-reflexive spaces is not stable for quotients by closed subspaces. When subspaces are concerned, one obtains

Proposition 2.15. If (E, t) is a b-reflexive lcs, then each closed subspace is b-reflexive in the relative topology and each subspace which is not closed is not b-reflexive. Proof. An lcs (E, t) is b-reflexive if and only if TE0 = τ(E0,E) holds. Thus, for a closed subspace (F, t|F ) we have to prove that TF 0 = τ(F 0,F ). Using the Corollary of the Proposition I.6.13, we obtain that τ(F 0,F ) = τ(E0,E)/F ◦ and then it follows that TF 0 = τ(F 0,F ). t If F = E , then obviously TE0 = TF 0 and (E0,TE0)0 = (F 0,TF 0)0 = E 6= F , and so a dense subspace F of a b-reflexive space is never b-reflexive. The same conclusion holds for arbitrary subspaces which are not closed, since they are dense in their closures. Since, as we have seen, there exists an even Fréchet-Montel space which has a nonreflexive quotient, A. Grothendieck defined in [41] the class of totally reflexive spaces. Namely, an lcs (E, t) is called totally reflexive if all of its quotients with respect to arbitrary closed subspaces F are reflexive. This, of course, does not mean that the strong topologies β(E0,E) and β(F ◦, E/F ) coincide on the space F ◦ =∼ (E/F )0 , but just that the respective duals coincide with the space E/F , for each closed subspace F of E . M. Valdivia recently characterized totally reflexive Fréchet space as a space isomorphic to a closed subspace of a product of countably many reflexive Banach spaces [123]. We give a proposition on subspaces and quotients of totally reflexive Fréchet spaces.

Proposition 2.16. The class of totally reflexive Fréchetlcs’s is stable with respect to each closed subspace and each quotient by a closed subspace. Proof. Let (E, t) be a totally reflexive (in particular reflexive) Fréchet lcs and let F be its t-closed subspace. If (H, (t/F )|H) is a closed subspace of the quotient space (E/F, t/F ), then the “quotient of the quotient” ((E/F )/H, (t/F )/H) is isomorphic with the space (E/G, t/G) for some t-closed subspace G of E , and so it is a reflexive lcs since (E, t) is totally reflexive. 56 II. Subspaces and quotients of topological vector spaces

Let now P be a t|F -closed subspace of F . It has to be shown that (F/P, (t|F )/P ) is a reflexive space. But, it is a closed subspace of (E/P, t/P ) and so it is semi-reflexive since (E, t) is totally reflexive. As a closed subspace of (E, t) it is also a Fréchet space, hence it is reflexive, too.

2.3. SUBSPACES OF TOPOLOGICAL VECTOR SPACES

As with locally convex spaces, the basic classes of topological vector spaces (ultrabarrelled, quasiultrabarrelled and ultrabornological spaces) are not stable with respect to arbitrary subspaces. An example can be constructed even in a simpler way than in the locally convex case. Take, e.g., an arbitrary vector space E with uncountable dimension, equipped with the finest locally convex topology tl . Then the space (E, tl) is a complete tvs which is not ultrabarrelled, but can be embedded as a closed subspace into a product of complete metrizable tvs’s which is ultrabarrelled. So, it is natural to ask whether the results about finite and countable codimensional subspaces from the previous paragraphs can be transfered to the non-locally convex situation. For deriving the results in this paragraph we shall often use the following ∗ notions. If (E, t) is a tvs, tb (resp. tb ; tβ ) will denote the linear topology on E generated by all closed (resp. closed bornivorous; bornivorous) strings in (E, t). Obviously, the space (E, t) is ultrabarrelled (resp. quasiultrabarrelled; ∗ ultrabornological) if and only if t = tb (resp. t = tb ; t = tβ ). Besides, the ∗ bounded sets in (E, t),(E, tb ) and (E, tβ) are the same. Theorem 3.1. (N. Adasch, B. Ernst [1]) Let (F, t|F ) be a finite-codimensional subspace of a tvs (E, t). If V = (Vn) is an ultrabarrel in (F, t|F ), then there exists an ultrabarrel U = (Un) in (E, t) such that Un ∩ F ⊂ Vn for all n ∈ N. Proof. It suffices to consider the case when F is a hyperplane in E . There are two possible cases:

◦ t ◦ t 1 Vn 6⊂ F for all n ∈ N; 2 Vn ⊂ F for some n0 (and all n > n0 ).

◦ t 1 In this case Un = Vn are the knots of the ultrabarrel U = (Un) which satisﬁes the requirements. ◦ 2 The knots Vn , n > n0 are closed in (E, t). Choose x0 ∈ E \ F and denote 1 by C the circled cover of {x }. Since C is compact, the sets U = V + C 0 n n0+n 2n are closed and absorbing in (E, t), so U = (Un) is an ultrabarrel that we have looked for. Corollary. (S. O. Iyahen [46]) Let (F, t|F ) be a ﬁnite-codimensional subspace of a tvs (E, t). Then topology tb induces the topology (t|F )b on F . In particular, if (E, t) is ultrabarrelled, then (F, t|F ) is ultrabarrelled, too. 2.3. Subspaces of topological vector spaces 57

In order to derive the corresponding result for ultrabornological spaces we shall introduce some new notions. Let (E, t) be a tvs. A sequence (xn) in E is a local zero-sequence if there exists a nondecreasing divergent sequence of positive scalars λn such that λnxn → 0 in (E, t).(xn) converges locally to x ∈ E if (xn − x) is a local zero-sequence. A linear mapping between tvs’s is locally continuous if it maps each local zero-sequence into a local zero-sequence. It is locally bounded if it maps each bounded set into a bounded set. It is easy to prove that a linear mapping is locally bounded if and only if it is locally continuous, if and only if it maps each local zero-sequence to a zero-sequence and if and only if it maps each local zero- sequence into a bounded set. In particular, the space (E, t) is ultrabornological if and only if each locally continuous mapping from E into an arbitrary (F)-space F is continuous.

Further, a sequence (xn) is a local Cauchy sequence if there exist positive scalars λn,m , n, m ∈ N such that λn,m ↑ ∞, n, m → ∞ and λn,m(xn − xm) → 0. It is easy to show that a locally continuous linear mapping maps each local Cauchy sequence into a Cauchy sequence. Finally, a subset M of (E, t) is said to be locally dense if each point in E is a limit of some locally convergent sequence from M . The following Lemma 3.2 was proved in [1]*, and Lemma 3.3 was (in the locally convex case) proved in [61].

Lemma 3.2. Let (E, t) be a tvs and F a locally dense hyperplane in (E, t). If f is a locally continuous linear mapping from (F, t|F ) into an (F)-space (G, t0), then there exists a locally continuous extension f˜ of f to the whole E .

Lemma 3.3. If (E, t) is an ultrabornological tvs and (F, t|F ) a hyperplane in E which is not locally dense. Then F is closed in E .

Theorem 3.4. (N. Adasch, B. Ernst [1]) If (E, t) is an ultrabornological tvs and (F, t|F ) its subspace of ﬁnite codimension, then (F, t|F ) is also an ultrabornological tvs. Proof. It is enough to consider the case codim F = 1. Two cases are possible: 1◦ F is locally dense in (E, t); 2◦ F is locally closed in (E, t). In the ﬁrst case the assertion follows from the Lemma 3.2 and the remarks before it. In the case 2◦ , by the Lemma 3.3, F is closed in (E, t). This means that (F, t|F ) is isomoprphic to the quotient of (E, t) by a closed subspace, and so it is ultrabornological.

Corollary. If (F, t|F ) is a ﬁnite-codimensional subspace of a tvs (E, t), then tβ|F = (t|F )β .

Theorem 3.5. (N. Adasch, B. Ernst [1]) Let (F, t|F ) be a ﬁnite-codimensional subspace of a tvs (E, t). If V = (Vn) is a bornivorous ultrabarrel in (F, t|F ), then there exists a bornivorous ultrabarrel U = (Un) in (E, t) such that Un ∩ F ⊂ Vn for all n ∈ N.

*See also an improved version in [56]. 58 II. Subspaces and quotients of topological vector spaces

Proof. We shall distinguish two cases — when the subspace F is dense in (E, tβ) and when it is closed in it (recall that tβ is the associated ultrabornological β t β topology). In the ﬁrst case, by the preceding corollary, Vn is a t -neighbourhood t tβ of the origin for each n and because of Un := Vn ⊃ Vn , U = (Un) is the desired bornivorous ultrabarrel in (E, t). Suppose now that F is closed in (E, tβ). Consider two more possibilities: ◦ t 1 There exists n0 ∈ N such that Vn = Vn for each n > n0 . Let x0 ∈ E \ F 1 and C be the circled cover of {x }. Then the sets U := V + C form a 0 n n0+n 2n bornivorous ultrabarrel U = (Un) in (E, t), which satisfy Un ∩ F ⊂ Vn for all n ∈ N. ◦ t t 2 For all n ∈ N, Vn 6= Vn . Then each Vn is absorbing in E . Let x0 ∈ E \F t and choose a decreasing sequence (λn) of positive scalars such that λnx0 ∈ Vn for λ each n ∈ N. Then the sets U 0 := V + n C (C is the circled cover of {x } as n n 2n 0 0 t before) form a bornivorous string in (E, t). Since Un ⊂ Vn−1 , the sets Un := Vn form the bornivorous ultrabarrel we were looking for.

Corollary. The class of quasiultrabarrelled spaces is stable with respect to ﬁnite-codimensional subspaces.

Parallel to countably barrelled and countably quasibarrelled lcs’s, countably ultrabarrelled and countably quasiultrabarrelled tvs’s were introduced in [47]. A σ -ultrabarrel (resp. bornivorous σ -ultrabarrel) in a tvs (E, t) is a (bornivorous) j ∞ ultrabarrel U = (U )j=0 which is a countable intersection of closed topological j T∞ j j ∞ strings, i.e. U = n=1 Un , j = 0, 1, 2,... , where Un = (Un)j=0 are closed topological strings for n = 1, 2,... . The space (E, t) is called countably ultrabarrelled (resp. countably quasiultrabarrelled) if every σ -ultrabarrel (resp. bornivorous σ - ultrabarrel) in it is topological. It is called an ultra-(DF) space if it is countably quasiultrabarrelled and if it has a fundamental sequence of bounded sets. Similarly as for other mentioned classes of tvs’s, the introduced classes of spaces are not stable for passing to arbitrary subspaces.

Theorem 3.6. [49], [83] Let (F, t|F ) be a subspace of ﬁnite codimension of ¡T ¢∞ a tvs (E, t) and V = (V j)∞ = ∞ V j a (bornivorous) σ -ultrabarrel in j=0 n=1 n j=0 ¡ ¢ j ∞ T∞ j ∞ (F, t|F ). There exists a (bornivorous) σ -ultrabarrel U = (U )j=0 = n=1 Un j=0 in (E, t), such that U j ∩ F = V j is valid for each j = 0, 1, 2,... . Proof. We shall prove the theorem for bornivorous σ -ultrabarrels; for σ - ultrabarrels it can be done using topologies tb and (t|F )b instead of tβ and (t|F )β . j ∞ It is enough to consider the case of F being a hyperplane in E . If (V )j=0 is the given bornivorous σ -barrel in (F, t|F ), then for each j = 0, 1, 2,... , V j = T∞ j β n=1 Vn is a (t|F ) -neighbourhood of the origin. Two cases are possible: 2.3. Subspaces of topological vector spaces 59

1◦ (F, (t|F )β) is dense in (E, tβ). It follows then that (F, t|F ) is dense in (E, t). Since, for each j = 0, 1, 2,... ,

t tβ ∞ ∞ ∞ \ t \ \ j j j Vn ⊃ Vn ⊃ Vn , n=1 n=1 n=1

¡ t¢ j ∞ T∞ j ∞ it follows that (U )j=0 = n=1 Vn j=0 is a bornivorous σ -ultrabarrel in (E, t) such that U j ∩ F = V j . ◦ β β T∞ j j β 2 (F, (t|F ) ) is closed in (E, t ). Then n=1(Vn + C) ⊃ V + C is a t - neighbourhood of the origin for each j = 0, 1, 2,... , where C is the circled cover of the set {x0}, x0 arbitrary chosen¡ from E \ F . If¢ (F, t|F ) is a closed subspace in j ∞ T∞ j (E, t), it is obvious that (U )j=0 = n=1(Vn +C) is a bornivorous σ -ultrabarrel in (E, t) such that U j ∩F = V j . If (F, t|F ) is a dense subspace in (E, t), then two t T∞ j more cases are possible: for each j = 0, 1, 2,... , n=1 Vn 6⊂ F or for some j0 and t ¡ t¢ T∞ j j ∞ T∞ j ∞ each j > j0 , n=1 Vn ⊂ F . In the ﬁrst case we take (U )j=0 = n=1 Vn j=0 t j T∞ j and in the second U = n=1 Vn for 0 6 j < j0 and

∞ \ ¡ t ¢ j j j0−j U = Vn + 2 C for j > j0 . n=1

j ∞ Then U = (U )j=0 is a bornivorous σ -ultrabarrel in (E, t) such that U ∩ F = V . This completes the proof of the theorem. Corollary. The classes of countably ultrabarrelled, countably quasiultrabarrelled and ultra-(DF) spaces are stable with respect to subspaces of finite codimension. At the end of this paragraph we give a result on countable infinite codimensional subspaces. Theorem 3.7. (N. Adasch, B. Ernst, D. Keim [3]) If (E, t) is an ultrabarrelled tvs and (F, t|F ) its subspace of countable infinite codimension, then (F, t|F ) is also an ultrabarrelled tvs.

Proof. Let (xn) be a cobasis of F in E ; denote F1 := F ⊕ span{x1}, F2 := F1 ⊕span{x2}, ... . If U = (Uj) is an ultrabarrel in (F, t|F ), then by the Theorem 1 1 ∞ 1 1.1 there exists an ultrabarrel U = (Uj )j=0 in (F1, t|F1) such that U = U ∩ F ; further, there is an ultrabarrel U 2 = (U 2)∞ in (F , t|F ) such that U 1 = U 2 ∩ F ¡ ¢ j j=0 2 2 ¡ ¢ 1 S∞ n ∞ S∞ n ∞ and so on. Now V = n=1 Uj j=0 is a string in (E, t) and n=1 Uj j=0 is topological since (E, t) is ultrabarrelled. We shall prove that

[∞ [∞ n n (1) Uj+1 ⊂ Uj , j = 0, 1,..., n=1 n=1 60 II. Subspaces and quotients of topological vector spaces

¡ ¢ S∞ n ∞ wherefrom it will follow that n=1 Uj j=0 is a topological string in (E, t) and S∞ n that the string U is topological in (F, t|F ), since Uj = n=1 Uj ∩F , j = 0, 1,... . S∞ n In order to prove the relation (1), let x∈ / n=1 Uj . Then there is a sequence (Vn) of topological strings Vn = (V n)∞ in (E, t) such that x∈ / U n + V n for T j j=0 j 1 all n. If we set W := ∞ U n + V n , then the string (W )∞ is topological k n=1 j+k 1+k ¡ k k=0¢ S∞ n since (E, t) is (countably) ultrabarrelled. But x∈ / n=1 Uj+1 + W1 , and so S∞ n x∈ / n=1 Uj+1 , which ﬁnishes the proof.

2.4. THREE-SPACE-PROBLEM FOR TOPOLOGICAL VECTOR SPACES

Of the basic classes of locally convex spaces described in the paragraph 2.1, only the classes of barrelled and of Mackey spaces are three-space stable (i.e. the three- space-problem (3SP) has the positive answer). In the class of topological vector spaces the property of “being ultrabarrelled” is 3SP-stable. Generally speaking, few properties of tvs’s are 3SP-stable. E.g., N. Kalton [55] and M. Ribe [89] showed that the property of “being locally convex” is not 3SP-stable in the class of tvs’s. For proving the three-space stability of some properties, the following result is often used: Lemma 4.1. (S. Dierolf, U. Schwanengel [33]) Two comparable vector topologies t1 and t2 on a vector space E are equal if and only if t1|F = t2|F and t1/F = t2/F hold for a certain subspace F .

Proof. Let, for example, t1 6 t2 and let U be a t2 -neighbourhood of the origin in E . Then there exists a t1 -neighbourhood of the origin V such that (V + V ) ∩ F ⊂ U . Since U ∩ V is a t2 -neighbourhood of the origin and since t1/F = t2/F , there exist t1 -neighbourhoods of the origin W1 and W2 such that W1 ⊂ V and W2 ⊂ (U ∩ V ) + F = q(U ∩ V )(q is the quotient mapping). Now ◦ W = W1 ∩ W2 is a t1 -neighbourhood of the origin which satisﬁes: 1 W ⊂ V and 2◦ W ⊂ (U ∩ V ) + F . If w ∈ W , then there exist x ∈ U ∩ V and y ∈ F such that w = x + y , whence y = w − x ∈ (W − (U ∩ V )) ∩ F ⊂ (V + V ) ∩ F ⊂ U . Thus, w = x + y ∈ U ∩ V + U ⊂ U + U , which proves that W ⊂ U + U . Hence, U + U is a t1 -neighbourhood of the origin and so t2 6 t1 . Proposition 4.2. (W. Roelcke, S. Dierolf [95]) If in the short exact sequence (∗), p. 39, of locally convex spaces the terms (F, t|F ) and (E/F, t/F ) are endowed with the weak (resp. Mackey) topologies, then the middle term (E, t) has the same property. Proof. The relations σ(F,F 0) = σ(E,E0)|F 6 t|F = σ(F,F 0), t/F > σ(E,E0)/F = σ(E/F, F ◦) = t/F 2.4. Three-space-problem for topological vector spaces 61 and the previous Lemma imply the equality of the comparable locally convex topologies σ(E,E0) and t on the vector space E . For the equalities σ(F,F 0) = σ(E,E0)|F and σ(E/F, F ◦) = σ(E,E0)/F see the end of the paragraph 1.6. The equality t = τ(E,E0) also follows from the mentioned Lemma and the fact that the quotient of a Mackey space is again a Mackey space. Indeed, τ(F,F 0) = t|F 6 τ(E,E0)|F 6 τ(F,F 0) τ(E/F, F ◦) = t/F 6 τ(E,E0)/F = τ(E/F, F ◦).

Remark. The 3SP-stability of the minimal, the finest locally convex and the finest linear topologies can be proved in a similar way, see [51]. Proposition 4.3. (W. Roelcke, S. Dierolf [95]) The class of barrelled lcs’s (resp. ultrabarrelled tvs’s) is 3SP-stable. Proof. Let T be a barrel in an lcs (E, t) from the short exact sequence (∗) in which (F, t|F ) and (E/F, t/F ) are barrelled spaces. Then there exists an absolutely convex and closed t-neighbourhood of the origin U such that (3U)∩F ⊂ t/F T ∩ F ⊂ T . Moreover, q(T ∩ U) is a barrel in the space (E/F, t/F ). Since this space is barrelled, the last set is a t/F -neighbourhood of the origin , and t/F t/F t because of q(T ∩ U) ⊂ q(T ∩ U + F ) , the set V := U ∩ T ∩ U + F is a t-neighbourhood of the origin. Now from the inclusion \ V ⊂ U ∩ (W ∩ U + T ∩ U + F ), W ∈U where U is a base of neighbourhoods of the origin in (E, t), and the condition (3U) ∩ F ⊂ T it follows \ t V ⊂ U ∩ (W ∩ U + T ∩ U + (3U) ∩ F ) ⊂ T ∩ U + T ⊂ 2T. W ∈U This proves that the barrel T is a t-neighbourhood of the origin. Thus we have proved that the middle term (E, t) in the sequence (∗) is a barrelled lcs. The proof for ultrabarrelled tvs’s is similar, see [95]. For constructing examples showing that certain properties are not 3SP-stable, the following Lemma, which is also of interest on itself, can be useful. Lemma 4.4. (S. Dierolf [30]) Let (E, t) be a tvs, (F, t|F ) its subspace and t1 a linear topology on the quotient space E/F which is finer than t/F . Let τ be the initial topology on E with respect to the mappings id: E → (E, t) and q : E → (E/F, t/F ) (i.e. the coarsest topology on E such that these mappings are continuous). Then τ > t, t|F = τ|F and τ/F = t1 . If the topologies t and t1 are locally convex, then τ is locally convex, too. Remark. For the proof of the Lemma the following observation is used: The set −1 { U ∩ q (V ) | U ∈ U(E, t),V ∈ U(E/F, t1) } 62 II. Subspaces and quotients of topological vector spaces is obviously a base of neighbourhoods of the origin for the linear topology τ on E satisfying the desired conditions. Moreover, τ is the coarsest linear topology on E with such properties. Then it is said that the pair ((E, t), (E/F, t1)) satisfies the lifting property. Similarly the pair ((E, t), (F, t0)) which satisfies the extension property is defined. For details see [30] (1), (2) and [95]. Now we are in position to give an example which shows that the classes of quasibarrelled lcs’s and quasiultrabarrelled tvs’s are not 3SP-stable. Example. Let (E, t) be a barrelled lcs and (F, t|F ) its dense subspace of countable infinite codimension such that all t-bounded subsets of E have finite- dimensional linear spans [5]. Applying the lifting property from the previous Lemma to the pair ((E, t), (E/F, k k)), where k k is an arbitrary fixed norm on E/F , we obtain: 1◦ (F, τ|F ) is a Montel space; 2◦ (E/F, τ/F ) is a normed, i.e. a quasibarrelled space; 3◦ (E, τ) is not countably quasibarrelled, where τ is the topology described in the Lemma. In fact, if (E, τ) were countably quasibarrelled, then it would also be countably barrelled, because from τ > t it follows that weakly and strongly bounded subsets in (E, τ) are the same. But then (E/F, τ/F ) would also be countably barrelled, and so also barrelled since it is normed. However, E/F is of countable infinite dimension and so it cannot be metrizable and barrelled. Contradiction. Remark. The previous example also shows that the class of countably quasibarrelled lcs’s (resp. countably quasiultrabarrelled tvs’s) is not 3SP-stable. The classes of ultrabornologique and bornological lcs’s (resp. ultrabornological tvs’s) are not 3SP-stable, too. For examples see [95], Lemma 1.4 and Example 2.15. Proposition 4.5. (W. Roelcke, S. Dierolf [95]) The class of countably barrelled lcs’s (resp. countably ultrabarrelled tvs’s) is 3SP-stable. Proof. Let the terms (F, t|F ) and (E/F, t/F ) in the sequence (∗) be countably T∞ barrelled lcs’s and let T = n=1 Un be a σ -barrel in the space (E, t). Using essentially the same technique as in the proof of the Proposition 4.3, one obtaims that V ⊂ 2T , where the t-neighbourhood of the origin V is given by V := U ∩ T t n∈N U ∩ Un + F , U is a t-neighbourhood of the origin such that (3U) ∩ F ⊂ T ∩ F ⊂ T . In the case of countably ultrabarrelled tvs’s there are no essential changes. Proposition 4.6. (W. Roelcke, S. Dierolf [95]) The class of (DF) lcs’s (resp. ultra-(DF) tvs’s) is not 3SP-stable. Proof. Let Y be an infinite-dimensional Fréchet-Montel space admitting continuous norms and let (E, τ(E,Y )) be the corresponding Mackey space, where E = Y 0 . Then (E, τ(E,Y )) is a nonnormed complete bornological (DF)-space containing a total bounded subset. Let now (Bn)n∈N be a fundamental sequence 2.4. Three-space-problem for topological vector spaces 63

of bounded subsets in the space (E, τ(E,Y )), such that linear span of B1 is τ(E,Y )-dense. Since span Bn 6= E holds for each n ∈ N, we can assume that span Bn $ span Bn+1 , n ∈ N. Now it is easy to construct a linear subspace F of infinite countable codimension in E such that dim(F + span Bn)/F < ∞ for each n ∈ N. Then by the Theorems 1.9 and 1.12, (F, τ(E,Y )|F ) is a barrelled, bornological (DF)-space. The lifting property from the Lemma 4.4, applied to the pair ((E, τ(E,Y )), (E/F, k k)), where k k is an arbitrary norm on E/F , gives that: 1◦ (F, τ|F ) is a barrelled, bornological (DF)-space; 2◦ (E/F, τ/F ) is a normed space; 3◦ (E, τ) is not a (DF)-space. In fact, suppose that (E, τ) is a (DF)-space and t/F let B be a bounded τ/F -neighbourhood of the origin. Then B ⊂ q(Bn) for some n ∈ N, wherefrom it follows that span(q(Bn)) is a dense subset in (E/F, τ/F ). But then F + span Bn is dense in the space (E, τ), which is impossible since F is a closed subspace in (E, τ) and dim(F + span Bn)/F is finite. Contradiction. As we have seen, few of the properties of lcs’s (tvs’s) are three-space-stable. However, sometimes additional conditions can assure that a certain property P is inherited from (F, t|F ) and (E/F, t/F ) to the middle term (E, t) of the short exact sequence (∗). A special type of these conditions will be described in the sequel. Let (E, t) and (G, s) be tvs’s and f : E → G a linear mapping. If for each s-bounded subset B there exists a t-bounded subset A such that B ⊂ f(A) (resp. B ⊂ f(A)), then it is said that f lifts (resp. lifts with closure) the bounded subsets of (G, s). Similarly, “lifting” of other types of bounded sets (e.g. Banach discs) can be defined. We already mentioned this condition in connection with quotient mappings in the first Chapter and we shall discuss this notion in more details in the next paragraph where we shall see how it turns out to be useful when dealing with Fréchet spaces. Here we just remark that using this notion, W. Roelcke and S. Dierolf proved [95] some results on the 3SP-stability. Indeed, suppose that if in the short exact sequence

q (∗) 0 → (F, t|F ) →i (E, t) → (E/F, t/F ) → 0 of lcs’s (tvs’s) the quotient mapping q lifts bounded sets with closure. Then it is easy to prove that if the spaces (F, t|F ) and (E/F, t/F ) are bornological, quasibarrelled or countably quasibarrelled (resp. ultrabornological, quasiultrabarrelled, countably quasiultrabarrelled), then the space (E, t) possesses the same property. We shall now adopt their method to prove similar results for some other classes of spaces. Recall (paragraph 2.2) that an lcs (E, t) is said to be a b-space (resp. b- barrelled) if every absolutely convex subset (resp. barrel) in it is a neighbourhood of the origin whenever its intersection with each bounded absolutely convex subset B of (E, t) is a neighbourhood of the origin in B (in the induced topology); (E, t) is a Db -space if it is b-barrelled and has a fundamental sequence of bounded subsets (equivalently, if it is a b-space with a fundamental sequence of bounded subsets). 64 II. Subspaces and quotients of topological vector spaces

The respective notions in the category of topological vector spaces are called locally topological, ultra-b-barrelled, resp. σ -locally topological spaces — in the previous deﬁnitions one has just to replace an absolutely convex set by a string and a barrel by an ultrabarrel [2], [50]. These properties of lcs’s (tvs’s) are not three-space-stable. As an example, let us show it for Db (resp. σ -locally topological) spaces. We shall use the example which was given in the proof of the Proposition 4.6. In it, (E, τ) is an lcs and (F, τ|F ) is its subspace, such that: ◦ 1 (F, τ|F ) is a barrelled and bornological (DF)-space; it is certainly a Db and σ -locally topological space; 2◦ (E/F, τ/F ) is a normed space; ◦ 3 (E, τ) is not a Db (neither a σ -locally topological) space. Indeed, the oposite assumption would lead to the contradiction in the same way as in the proof of Proposition 4.6, using the fact that the quotient mapping q : E → E/F in this case lifts bounded sets with closure — see Theorem 3.2.1 [76], resp. 4.(3) [2]. To show that the three-space-problem for the mentioned classes has the positive answer under some additional hypothesis let us prove ﬁrst the following

Lemma 4.7. [95] Let in the short exact sequence (∗) of tvs’s the space (F, t|F ) be locally bounded (i.e. possesses a bounded neighbourhood of the origin). Then the quotient mapping q lifts bounded subsets. Moreover, if C is a t/F -bounded subset, then there exists a t-bounded subset B such that C = q(B). Proof. Let U be a circled and open t-neighbourhood of the origin such that F ∩ (U + U) is t|F -bounded. Since q is open, there exists n ∈ N such that C ⊂ nq(U). The set B := nU ∩ q−1(C) clearly satisﬁes q(B) = C . To show that B is t-bounded, let W be a circled t-neighbourhood of the origin. Then there is m ∈ N such that F ∩ (U + U) ⊂ mW and there is k ∈ N, k > n such that C ⊂ kq(W ∩ U). From B ⊂ nU ∩ (k(W ∩ U) + F ) we get B ⊂ nU ∩ [k(W ∩ U) + F ∩ (nU + kU)] ⊂ kW + F ∩ k(U + U) ⊂ kW + kmW ⊂ km(W + W ). This proves that B is t-bounded.

Proposition 4.8 [51] (a) Let (∗) be a short exact sequence of tvs’s, such that (F, t|F ) is locally bounded and (E/F, t/F ) is locally topological. Then (E, t) is a locally topological space. (b) If the previous sequence is in the category of lcs’s, such that (F, t|F ) is seminormed and (E/F, t/F ) is a b-space, then (E, t) is a b-space, too. Proof. We shall prove the part (a); for (b) it can be done in an even simpler way. ∞ Let (Tn)1 be a locally topological string in (E, t), i.e. let for each B ∈ B(E) (the set of all absolutely convex bounded subsets of (E, t)) and for each n, Tn ∩ B ∞ be a neighbourhood of the origin in B in the induced topology. Then (Tn ∩ F )1 is 2.5. Three-space-problem in Fr´echet lcs’s 65 a locally topological string in (F, t|F ). As far as the space (F, t|F ), being locally ∞ bounded, is locally topological, (Tn ∩ F )1 is a topological string, i.e. there exists ∞ a topological string (Un)1 in (E, t), such that F ∩ (Un + Un) ⊂ Tn for each n. It can be assumed that the sets F ∩ (Un + Un) are bounded. Consider the string ∞ (q(Tn ∩ Un))1 in E/F and let us show that it is locally topological. Let C be an arbitrary bounded subset of (E/F, t/F ). According to the previous Lemma, there is B ∈ B(E, t), such that C = q(B). It follows that

C ∩ q(Tn ∩ Un) = q(B) ∩ q(Tn ∩ Un) ⊃ q(B ∩ Tn ∩ Un), hence, C ∩ q(Tn ∩ Un) is a neighbourhood of the origin in C , since q is an open ∞ mapping. Therefore, (q(Tn∩Un))1 is a t/F -topological string in E/F , and so, e.g., q(T2 ∩ U2) is a neighbourhood of the origin in E/F . Now V = U2 ∩ ((T2 ∩ U2) + F ) is a t-neighbourhood of the origin in E . Since

V ⊂ U2 ∩ (T2 ∩ U2 + F ∩ (U2 + U2)) ⊂ T2 + T2 ⊂ T1,

T1 is a neighbourhood of the origin in E , too, which proves that the space (E, t) is locally topological. In the similar way one can prove the following Proposition 4.9. [51] Let (∗) be a short exact sequence of lcs’s (tvs’s) in which (F, t|F ) is seminormed (locally bounded) and (E/F, t/F ) is (ultra) b- barrelled. Then (E, t) is (ultra) b-barrelled, too. Proposition 4.10. [51] Let (∗) be a short exact sequence of lcs’s (tvs’s) in which (F, t|F ) is seminormed (locally bounded) and (E/F, t/F ) is a Db - (resp. σ - locally topological) space. Then (E, t) is a Db - (resp. σ -locally topological) space, too. Proof. Follows by combining our Proposition 4.8 and Prop. 3.3(a) of [95]. Remark. Besides the properties we have proved in this paragraph to be 3SP-stable, some other classes of lcs’s, like complete, nuclear, Schwartz [95] and metrizable spaces [40] have this property. We do not know whether σ -barrelled and B - and Br -complete spaces of Pt´ak[63], [103] are 3SP-stable.

2.5. THREE-SPACE-PROBLEM IN FRECHET´ LCS ’S

If all of the terms of the short exact sequence q (∗) 0 → (F, t|F ) →i (E, t) → (E/F, t/F ) → 0 belong to some narrower class of locally convex spaces, then sometimes the three- space-problem can have the positive answer, even it is negative in general. For example [95], the classes of Montel (see further the Proposition 5.3), reflexive and quasinormed spaces among the (F)-spaces are 3SP-stable. Moreover, the class of 66 II. Subspaces and quotients of topological vector spaces semi-reflexive spaces among the (F)- or (DF)-spaces is 3SP-stable [95], see further the Proposition 5.4. In the class of (F)-spaces 3SP-stability holds also for the properties (DN) and (Ω) (D. Vogt, [124] and [125]) and the same is true for the so-called quojections [71]. A Fréchet space (E, t) is distinguished (resp. has the property (DC)*) if its strong dual is bornological [equivalently, barrelled, quasibarrelled or ultrabornologique] (resp. if β(E0,E)-bounded subsets of its dual E0 are metrizable). In the class of (F)-spaces neither of these properties is 3SP-stable (J. Bonet, S. Dierolf, C. Fernandez [17]). A Fréchet space (E, t) is a dual space if it is isomorphic to the strong dual of some barrelled (DF)-space (X, p). Similarly, the short exact sequence j q 0 → (F, r) → (E, s) → (G, t) → 0 of (F)-spaces is called a dual sequence if there exists a short exact sequence i p 0 → (X, t1) → (Y, t2) → (Z, t3) → 0 of barrelled (DF)-spaces, such that the spaces (G, t),(E, s),(F, r) are respectively t t isomorphic to the strong duals of (X, t1),(Y, t2),(Z, t3) and j = p , q = i [29]. The property of “being a dual space” is not 3SP-stable in the class of Fréchet spaces, too. When proving that some properties are 3SP-stable, it is important to know the relationship between the topologies β(F 0,F ) and β(E0,E)/F ◦ , as well as β(F ◦, E/F ) and β(E0,E)|F ◦ in the topological duals (F, t|F )0 =∼ E0/F ◦ and (E/F, t/F )0 =∼ F ◦ . We shall prove the following two propositions. Proposition 5.1. If (F, t|F ) is a countably quasibarrelled subspace of a lcs (E, t) such that (F 0, β(F 0,F )) is bornological, then β(F 0,F ) = β(E0,E)/F ◦ . Proof. It has to be proved that the identity mapping id: (F 0, β(F 0,F )) → (F 0, β(E0,E)/F ◦) is continuous. Since the strong dual (F 0, β(F 0,F )) is bornologi- 0 cal, it is sufficient to prove that each β(F ,F )-bounded sequence { fn | n ∈ N } is β(E0,E)/F ◦ -bounded. Each such sequence is t|F -equicontinuous (because (F, t|F ) is countably quasibarrelled), and on the base of [62], 22.1.(1)b) there exists a 0 t t-equicontinuous subset A ⊂ E such that { fn | n ∈ N } ⊂ j (A), where jt : E0 → E0/F ◦ is the quotient mapping. Since obviously jt(A) is a β(E0,E)/F ◦ - bounded subset, then the sequence { fn | n ∈ N } is of the same kind. Proposition 5.2. The topologies β(F ◦, E/F ) and β(E0,E)|F ◦ are equal if and only if the quotient mapping q :(E, t) → (E/F, t/F ) lifts bounded subsets with closure. Corollary 1. If the space (F, t|F ) in the sequence (∗) is distinguished and q lifts bounded subsets with closure, then the sequence t t 0 q 0 j 0 (∗∗) 0 → (E/F )β → Eβ → Fβ → 0 is exact.

*Density condition of S. Heinrich [43] 2.5. Three-space-problem in Fr´echet lcs’s 67

Remark that from [70] it follows that for the exactness of the previous sequence it is enough to assume that q lifts bounded subsets, see further the Proposition 5.9.

Corollary 2. (J. Bonet, S. Dierolf, C. Fernandez [17]) If q lifts bounded subsets with closure, then the 3SP for distinguished Fr´echet(resp. (F)-spaces with the condition (DC)) has the positive answer. Proof. In fact, the exactness of the sequence (∗∗) and the Proposition 4.3 imply that the middle term is barrelled and so the space (E, t) is distinguished. On the base of [11], Theorem 2 it follows that the middle term (E, t) has the ∞ 0 property (DC) if and only if l (Eβ) is a barrelled/bornological space. Since each (F)-space with the property (DC) is distinguished, the sequence (∗∗) is an exact sequence of bornological (DF)-spaces. Thus the sequence

t t ∞ 0 qˆ ∞ 0 ˆi ∞ 0 0 → l ((E/F )β) → l (Eβ) → l (Fβ) → 0 is algebraically and also topologically exact (for details see [17], [11]). Since now the exterior terms are barrelled, such is the middle term, too, and the (F)-space (E, t) satisﬁes (DC).

Proposition 5.3. Among (F)-spaces the class of Montel spaces is 3SP-stable. Proof. Knowing that the exterior terms of the sequence (∗) are (FM)- spaces, we conclude that (∗∗) is a short exact sequence. Actually, by the Banach- Dieudonn´etheorem [62], 21.10, we have 0 ◦ 0 ◦ 0 ◦ Ec|F = (E/F )c = β(F , E/F ) = β(E ,E)|F , wherefrom it follows that q lifts bounded subsets with closure. As the space (F, t|F ) 0 0 0 ◦ 0 ◦ is Montel, it is distinguished and β(F ,F ) = Fc 6 Ec/F 6 β(E ,E)/F = 0 0 0 β(F ,F ) (according to the Proposition 5.1). Thus, by the Lemma 4.1, Ec = Eβ , i.e. (E, t) is a Montel space. Moreover, (∗) is a dual sequence. According to [95], Lemma 1.4 and Example 1.5, the classes of Montel and of semi-reﬂexive spaces are not 3SP-stable in general. But under additional assumptions we have

Proposition 5.4. (W. Roelcke, S. Dierolf [95]) If for a closed subspace (F, t|F ) in the sequence (∗) β(F 0,F ) = β(E0,E)/F ◦ holds, and if the exterior terms in this sequence are semi-reﬂexive, then the middle term (E, t) is semi- reﬂexive, too. Proof. The given condition and the relation σ(E0,E)|F ◦ = σ(F ◦, E/F ) 6 τ(E0,E)|F ◦ 6 β(E0,E)|F ◦ 6 β(F ◦, E/F ) = τ(F ◦, E/F ) imply the exactness of the sequence 0 → (F ◦, β(E0,E)|F ◦) → (E0, β(E0,E)) → (E0/F ◦, β(E0/F ◦,F )) → 0, 68 II. Subspaces and quotients of topological vector spaces i.e. of the sequence 0 → (F ◦, β(E0,E)|F ◦) → (E0, β(E0,E)) → (E0/F ◦, τ(E0/F ◦,F )) → 0. This implies the (algebraic) exactness of the sequences 0 → F → E → E/F → 0 0 → F → E00 → E/F → 0, where E00 = (E0, β(E0,E))0,F = (E0/F ◦, β(E0/F ◦,F ))0 and E/F = (F ◦, β(E0,E)|F ◦)0. Let us prove that E00 ⊂ E . If x00 ∈ E00 , then x00|F ◦ (restriction to F ◦ ) is an 00 0 0 element of E/F and so there exists x1 ∈ E such that x (x ) = x (x1) for all 0 ◦ 00 00 x ∈ F . Since x − x1 ∈ E , there exists a t-closed absolutely convex and t- 00 0 00 σ(E ,E ) ◦◦ bounded subset B of E such that x − x1 ∈ B = B (polars are taken 00 0 00 with respect to the pair hE ,E i). This implies that x − x1 is a bounded linear ◦ ◦ 00 ◦ 00 form on B + F (because x − x1|F = 0), i.e. x − x1 is a continuous linear form on the space (E0/F ◦, β(E0,E)/F ◦) = (E0/F ◦, β(E0/F ◦,F )) = (F 0, τ(F 0,F )), 00 0 0 0 0 so that there exists x2 ∈ F such that (x − x1)(x ) = x2(x ) for all x ∈ F . Thus, 00 x = x1 + x2 ∈ E + F ⊂ E + E = E .

Remark. Among (F) (resp. (DF), Db ) spaces the class of semi-reflexive spaces is 3SP-stable, since they obey the conditions of the previous Proposition. In [17] some counterexamples are given which show that without lifting of bounded subsets the properties of “being distinguished” and “having the (DC)” are not 3SP-stable. The method of constructing such examples is called sometimes “Pisier method” (see [31]). In the sequel we shall describe this method and some related results. Let X be the given Fréchet space, L its closed subspace and Z a Fréchet space, continuously embedded in the quotient X/L. The mapping p: X × Z → X/L, defined by p(x, z) = q(x) − z , where q : X → X/L is the quotient mapping, is linear, continuous and surjective, which means that it is a quotient mapping, too. Therefore we have the following short exact sequence: p 0 → Y := ker p ,→ X × Z → X/L → 0. The aim of such construction is that X , L and Z be chosen in such a way that X/L be a certain separable Banach space (Y then becomes Montel, see [31]), then that Z does not possess the given property so that X × Z also does not possess the property. If X is a Fréchet-Montel space, then Y is Montel if and only if each bounded sequence in Z which converges in X/L also converges in Z . This observation leads us to Proposition 5.5. (S. Dierolf [31]) Let P be a property in the class of (F)- spaces, such that it is obeyed by the space l1 and by each (FM)-space. Suppose that one of the following two conditions is satisfied: 2.5. Three-space-problem in Fréchet lcs’s 69

1◦ P implies distinguishedness; 2◦ each (F)-space with the property P is topologically complemented in its bidual. Then the property P is not 3SP-stable. Proof. See [31], p. 83. Corollary. (S. Dierolf [31]) The properties of being distinguished, (DC) or a dual space in the class of (F)-spaces are not 3SP-stable. Proof. Since each (FM)-space and l1 obeys the (DC) and since (DC) implies distinguishedness, we conclude that (DC) and distinguishedness itself are not 3SP- stable properties. As the strong dual of a quasibarrelled (DF)-space has the topological complement in its bidual and as (FM)-spaces and l1 are dual spaces, it follows that the property of “being a dual space” is also not 3SP-stable. By a delicate example it was shown in [31] that the notions “being a strong dual of a barrelled (DF)-space” and “being a strong dual of a (DF)-space” in the class of (F)-spaces are different. Taking in the described Pisier method X to be the Köthe-Grothendieck Fréchet-Montel space (see the Example before Proposition 2.14), it can be shown Proposition 5.6. There exists a short exact sequence 0 → (F, t|F ) → (E, t) → l1 → 0 such that (F, t|F ) is a Montel spaces and that (E, t) is not the strong dual of a (DF)-space. Thus, the property of “being the dual of a (DF)-space” is not 3SP- stable. Proof. See [29], Example 1 and [31], Corollary 4. The concept of lifting of bounded subsets (with or without closure) took the basic role in the work of majority of authors who considered the 3SP in the class of (F)-spaces. Also, in solving the Grothendieck’s problem: “is the bidual of a distinguished Fréchet space always distinguished?”, the concept of lifting of bounded sets with closure was cruical. We state the following proposition. Proposition 5.7. (S. Dierolf [31]) Let E be a Fréchetspace. Then the following conditions are equivalent: (a) The bidual E00 is distinguished. (b) The spaces E and E00/E are distinguished and the quotient mapping q : E00 → E00/E lifts bounded sets with closure. For the proof of this proposition we shall use the following two results from the duality theory of Fréchet spaces. Proposition 5.8. (J. Bonet, S. Dierolf, C. Fernandez [18]) Let E and F be Fréchetspaces such that E ⊂ F ⊂ E00 and let q : F → F/E be the quotient mapping. Suppose that F is a distinguished space. Then: 70 II. Subspaces and quotients of topological vector spaces

1◦ F/E is distinguished; 2◦ q lifts bounded subsets with closure. Proof. Let j : E → F and i: F → E00 be natural injections. Then the corresponding transposed mappings jt :(F 0, β(F 0,F )) → (E0, β(E0,E)) and it :(E000, β(E000,E00)) → (F 0, β(F 0,F )) are continuous. Since F is a distinguished Fr´echet space, i.e. since β(F 0,F ) = β(F 0,F 00), the mapping jt :(F 0, β(F 0,F )) → (E0, β(E0,E00)) is also continous. It follows from β(E000,E00)|E0 6 β(E0,E00) that the mapping it|E0 :(E0, β(E0,E00)) → 0 0 t t 0 0 (F , β(F ,F )) is continuous. Hence, the restriction j ◦i |E is the identity on E , wherefrom it follows that the space (F 0, β(F 0,F )) is the direct topological sum of the subspaces (it(E0), β(F 0,F )|it(E0)) and (E◦, β(F 0,F )|E◦), where E◦ = ker jt . Since (F 0, β(F 0,F )) is a barrelled space, (E◦, β(F 0,F )|E◦) has the same property. This and the fact that σ(E◦, F/E) = σ(F 0,F )|E◦ 6 β(F 0,F )|E◦ 6 β(E◦, F/E) imply that β(F 0,F )|E◦ = β(E◦, F/E)*, and so (E◦, β(E◦, F/E)) is a barrelled space. Thus, F/E is a distinguished Fr´echet space and the mapping q : F → F/E lifts bounded subsets with closure. Proposition 5.9. [70], 26.12 If the mapping q in the sequence (∗) lifts bounded subsets with closure, then 0 0 0 0 → (E/F )β → Eβ → Fβ → 0 is a short exact sequence. Proof of the Proposition 5.7. (a) =⇒ (b). Taking in the Proposition 5.8 F = E00 we obtain that E00/E is a distinguished space and that q : E00 → E00/E lifts bounded subsets with closure. Thence we have a short exact sequence q (1) 0 → E →i E00 → E00/E → 0. Using the Proposition 5.9 we conclude that the sequence 00 0 00 0 0 0 → (E /E)β → (E )β → Eβ → 0 0 00 0 000 is exact, i.e. Eβ is a barrelled space since (E )β = Eβ is of this kind. Hence, E is a distinguished space. (b) =⇒ (a). Applying Corollary 2 of the Proposition 5.2 to the sequence (1), we obtain that the space E00 is distinguished. Remark. As we have already said, three-space-stability of (DC)-spaces was investigated in [17]. In this context the following question was posed in [15]: is there an (F)-space E without (DC), such that its strong bidual E00 possesses (DC)? A. Peris in his dissertation showed that the answer was negative. We give here an alternative proof of this fact [51]. Let E be an (F)-space, such that E00 has (DC). Then E00 is a distinguished 000 (F)-space, such that bounded subsets of its strong dual Eβ are metrizable. Ac- cording to the Proposition 5.7, the spaces E and E00/E are distinguished and the

*If σ(E,E0) 6 t 6 β(E,E0) and (E, t) is a barrelled space, then t = β(E,E0). 2.5. Three-space-problem in Fr´echet lcs’s 71 quotient-map q : E00 → E00/E lifts bounded sets with closure (and also without closure — see further the Proposition 5.10). So we have a short exact sequence of (F)-spaces q 0 → E →i E00 → E00/E → 0 in which q lifts bounded sets. Applying the Proposition 5.9, the dual sequence

t t 00 0 q 00 0 i 0 0 → (E /E)β → (E )β → Eβ → 0 000 t is topologically exact. Since Eβ is a (DF)-space, quotient-map i lifts bounded 000 sets with closure. As far as bounded sets in Eβ are metrizable, bounded sets in 0 Eβ have the same property, and so the space E satisfies (DC). We finish this paragraph with one of the nicest between new results concerning 3SP of Fréchet spaces. Theorem 5.10. (J. Bonet, S. Dierolf [16]) Let (E, t) be a Fréchetspace and (F, t|F ) its closed subspace such that q :(E, t) → (E/F, t/F ) lifts bounded subsets with closure. Then q also lifts bounded subsets. This means that in all assertions on Fréchet spaces the condition of lifting of bounded subsets can be replaced by the condition of lifting of bounded subsets with closure. The following corollary shows how the distinguishedness of the quotient E/F can be expressed in the terms of the space (F ◦, β(E0,E)|F ◦). Corollary. [16] Let (E, t) be a Fréchetspace, (F, t|F ) its closed subspace and q :(E, t) → (E/F, t/F ) the quotient mapping. Then the following conditions are equivalent: (a) (F ◦, β(E0,E)|F ◦) is ultrabornologique; (b) (F ◦, β(E0,E)|F ◦) is bornological; (c) (F ◦, β(E0,E)|F ◦) is barrelled; (d) (E/F, t/F ) is distinguished and q lifts bounded subsets. Proof. (a) =⇒ (b) =⇒ (c) because (F ◦, β(E0,E)|F ◦) is a complete lcs. (c) =⇒ (d). Since β(F ◦, E/F ) > β(E0,E)|F ◦ > σ(F ◦, E/F ), we have β(F ◦, E/F ) = β(E0,E)|F ◦ because (F ◦, β(E0,E)|F ◦) is a barrelled space. Fur- ther, the Proposition 5.2 implies that the mapping q lifts bounded subsets with closure, which by the Theorem means that it lifts bounded subsets. (d) =⇒ (a). Since q lifts bounded subsets, it also lifts bounded subsets with closure, which means that β(E0,E)|F ◦ = β(F ◦, E/F ). The fact that (E/F, t/F ) is a distinguished Fréchet space implies that its strong dual (F ◦, β(F ◦, E/F )) = (F ◦, β(E0,E)|F ◦) is ultrabornologique. III. ORDERED TOPOLOGICAL VECTOR SPACES

3.1. BASICS OF THE THEORY OF RIESZ SPACES

Let E be a vector space over the field R of real numbers. A nonempty convex subset C of E is called a cone if λC ⊂ C for all λ > 0. Obviously, the cone C in the space E defines a reflexive and transitive relation “6” given by

x 6 y if and only if y − x ∈ C.

The relation so deﬁned agrees with the vector structure, i.e.

x > 0 and y > 0 =⇒ x + y > 0 x > 0 and λ > 0 =⇒ λx > 0 holds. Then the pair (E,C), i.e. (E, 6) is called an ordered vector space. Con- versely, each relation “6”, reflexive, transitive and compatible with the vector structure, defines a cone C in the vector space E by C = { x | x > 0 } and then the relation “6” is determined by the cone C . The cone C is called proper if C ∩ (−C) = {0}, i.e. if the relation “6” is antisymmetric. It is clear that each subspace F of the vector space E determines an order relation “6”, but it is not antisymmetric. Such orders are not considered. The cone C is generating in E if E = C − C holds. We give some important examples of ordered vector spaces. Examples. 1◦ Let X be a locally compact Hausdorff space and let C(X) be the space of all real-valued continuous functions with compact support. Order “6” in C(X) is defined by f 6 g if and only if f(x) 6 g(x) for all x ∈ X . This is one of the most important examples in the theory of integration. The positive cone in C(X) is proper and generating. ◦ 2 Let (X, µ) be a measurable space and p > 0. Then the space Lp(X, µ) of p-summable functions is an ordered vector space, where “6” is defined by: f 6 g if and only if f(x) 6 g(x) a.e., with respect to the measure µ. The positive cone is proper and generating. 3◦ The space C(k)[0, 1] of functions can be ordered in the natural way, i.e. as in the example 1◦ . The cone of positive elements in this example is proper and generating, too. 3.1. Basics of the theory of Riesz spaces 73

The following property is easy to prove. Proposition 1.1. Let (E,C) be an ordered vector space. Then the following conditions are equivalent: (a) C is a generating cone (E = C − C); (b) (∀x ∈ E)(∃c ∈ C) c > x; (c) (∀x ∈ E)(∀y ∈ E)(∃z ∈ E) x 6 z and y 6 z . If (E,C) is an ordered vector space and x, y ∈ E , x 6 y , then the set

[x, y] = { z ∈ E | x 6 z 6 y } = (x + C) ∩ (y − C) is called an order-interval. A subset A in the space (E,C) is order-bounded if it is contained in some order-interval. It is order-convex if [x, y] ⊂ A whenever x, y ∈ A and x 6 y . It has to be remarked that the families of convex and order- convex subsets of an ordered vector space are uncomparable. If A is an arbitrary subset in the space (E,C), then [A] := { z ∈ E | x 6 z 6 y for some x, y ∈ A } = (A+C)∩(A−C) is called the order-convex cover of A and it obviously contains A. S A subset A in an ordered vector space (E,C) is called solid if A = { [−a, a] | a ∈ A ∩ C }. The space (E,C) is said to possess the Riesz decomposition property if [0, x + y] = [0, x] + [0, y] for all x, y ∈ C . If, moreover, the cone C is generating, the space (E,C) is called a weak Riesz space. Definition 1. An ordered vector space (E,C) with a proper cone C is called a Riesz space (vector lattice) if each pair of elements x, y ∈ E has its supremum (the least upper bound) in E (denoted by x ∨ y or sup(x, y)). Of course, the previous condition is equivalent to: each pair of elements x, y ∈ E has its infimum (the greates lower bound) in E , denoted by x ∧ y or inf(x, y). By the equality sup(x, y) = sup(x − y, 0) + y , the ordered vector space (E,C) is Riesz if and only if for each x ∈ E there exists sup(x, 0). It is clear that in each Riesz space (E,C) the cone C is generating and Riesz decomposition property is satisfied. The following example shows that the converse is not true. Example. Let the space R2 be ordered by the cone C = { (x, y) ∈ R2 | x > 0, y > 0 } ∪ {(0, 0)}. It is easy to check that (R2,C) is a weak Riesz space, but not a Riesz space. Let now (E,C) be a Riesz space and x ∈ E . New elements in E are defined by x+ = sup(x, 0); x− = sup(−x, 0); |x| = sup(−x, x); they are called positive part, negative part and absolute value of x, respectively. Two elements x, y in (E,C) are called disjoint (x ⊥ y ) if inf(|x|, |y|) = 0. For an arbitrary subset A of E we put

Ad = { x ∈ E | x ⊥ a for all a ∈ A }. 74 III. Ordered topological vector spaces

In connection with the introduced notions the following properties can be easily proved (see, e.g., [137]).

Proposition 1.2. For arbitrary elements x and y of a Riesz space (E,C) the following holds: 1◦ x+ ⊥ x− ; 2◦ |x| = x+ + x− = sup(x+, x−); 3◦ x = x+−x− is the unique representation of x as a diﬀerence of two disjoint positive elements; 4◦ x + y = sup(x, y) + inf(x, y); 5◦ |x + y| 6 |x| + |y|; 6◦ |x| 6 y if and only if −y 6 x 6 y .

Solid subsets play an important role in the theory of topological Riesz spaces. We give a characterization of such subsets in terms of the lattice structure.

Proposition 1.3. A subset A in a Riesz space (E,C) is solid if and only if for each x, y ∈ E , |x| 6 |y| and y ∈ A =⇒ x ∈ A.

Proof. Let A be solid and let |x| 6 |y|, y ∈ A. It means that −a 6 y 6 a for some a ∈ A ∩ C . Thus, |x| 6 a, and so −a 6 x 6 a and x ∈ A sinceS A is solid. Conversely, if the condition is satisﬁed, we have to show that A = { [−u, u] | u ∈ A ∩ C }. The given condition impliesS that |a| ∈ A whenever a ∈ A. Then from −|a| 6 a 6 |a| it follows that A ⊂ { [−u, u] | 0 6 u ∈ A }. On the other hand, if ◦ −a 6 x 6 a for someS a ∈ A ∩ C , then we have |x| 6 a (Porposition 2.6 ) and so a ∈ A. Therefore, { [−u, u] | 0 6 u ∈ A } ⊂ A, i.e. A is solid. Now it is not hard to prove that an arbitrary intersection (union) of solid susbsets is either empty or again a solid subset. If A is a subset of a Riesz space (E,C), then the smallest solid subset which contains A (denotedS by SA ) is called the solid cover of A. It can be easily checked that SA = { [−|a|, |a|] | a ∈ A }. The solid cover of a singleton {x} is denoted by Sx , i.e. Sx = [−|x|, |x|]. The solid kernel of a set A (denoted by sk(A)) is deﬁned by

sk(A) = { x ∈ E | [−|x|, |x|] ⊂ A }.

It is either empty or the greatest solid subset of E contained in A and it can be also written as [ sk(A) = { [−u, u] | [−u, u] ⊂ A }.

Some properties of solid subsets are contained in the next proposition. 3.1. Basics of the theory of Riesz spaces 75

Proposition 1.4. Let (E,C) be a Riesz space and let A be a subset of E . Then the following is valid: 1◦ If A is solid, then its convex cover co A is solid; 2◦ If A is convex, than its solid kernel sk(A) is convex; 3◦ If sk(A) 6= ∅, then sk(A) is absorbing if and only if sk(A) absorbs all order-bounded subsets; 4◦ A absorbs all order-bounded subsets of E if and only if sk(A) absorbs all order-bounded subsets; 5◦ If A and B are solid subsets of E , then A + B and λA, λ ∈ R are also solid subsets of E . ◦ Proof. 1 Since Sλx = λSx for all λ ∈ R, we shall prove ﬁrst of all that 1 Sx+y ⊂ Sx +Sy . If z ∈ Sx+y , then |z| 6 |x+y| 6 |x|+|y|, i.e. 0 6 2 (z+|x|+|y|) 6 |x| + |y|. Using the Riesz decomposition property one obtains that 0 6 v1 6 |x|, 0 6 v2 6 |y| for some v1 , v2 ∈ E and z+|x|+|y| = 2v1 +2v2 . Taking u = 2v1 −|x|, w = 2v2 − |y|, we get u ∈ Sx , w ∈ Sy and z = u + w, which gives Sx+y ⊂ Sx + Sy .

If now A is a solid subset of E and x ∈ co A, then there exist ai ∈ A and Pn Pn λi ∈ [0, 1], λi = 1 such that x = λiai . Then we have i=1 i=1 Pn Pn Sx ⊂ λiSai ⊂ λiA ⊂ co A, i=1 i=1 which means that co A is solid. 2◦ We know that sk(A) is the greatest solid subset of E contained in A and by 1◦ , co(sk(A)) is solid. Now because of co(sk(A)) ⊂ A it follows that sk(A) = co(sk(A)), i.e. sk(A) is convex. 3◦ Since sk(A) is a solid subset, then sk(A) absorbs u ∈ C if and only if it absorbs the whole order-interval [−u, u]. 4◦ The condition is clearly suﬃcient since sk(A) ⊂ A. Conversely, we have to prove that sk(A) absorbs order-intervals [−u, u], u ∈ C if A does. From [−u, u] ⊂ λA for some λ > 0, it follows that [−u, u] ⊂ sk([−u, u]) ⊂ sk(λA) = λ sk(A), i.e. sk(A) absorbs order-intervals. 5◦ If B is a solid subset, then λB is solid, too. We have to check the solidity of the sum A + B if A and B are solid. Let x ∈ A, y ∈ B and let z ∈ E such that |z| 6 |x + y|. Then there exists a ∈ Sx , b ∈ Sy such that z = a + b (since Sx+y ⊂ Sx + Sy ). The solidity of A and B implies that Sx ⊂ A, Sy ⊂ B and thence z ∈ A + B , which by the Proposition 1.3 means that the sum A + B is solid. Example. Let R2 be ordered by the positive cone C = { (x, y) | x > 0, y > 0 }. Then (R2,C) is a Riesz space. Take A = { λ(−2, 0) + (1 − λ)(1, 3) | λ ∈ [0, 1] }. The set A is convex, but the solid cover SA of A is not convex, because the set { λ(−2, 0) + (1 − λ)(−1, 3) | λ ∈ [0, 1] } is not contained in SA . Thus, in general, the solid cover of a convex set need not be convex. 76 III. Ordered topological vector spaces

A vector subspace F of a Riesz space (E,C) is its Riesz subspace (sublattice) if sup(x, 0) ∈ F for each x ∈ F . Remark that (F,C ∩ F ) can be a Riesz space, while F is not a Riesz subspace of (E,C) ([103], Exercise V.14(d)). By an l-ideal in a Riesz space (E,C) we mean a solid subspace and by an l-homomorphism a linear mapping which “conserves” lattice operations. A Riesz subspace F of a Riesz space (E,C) is an l-ideal if and only if F is order-convex, i.e. F = (F + C) ∩ (F − C) holds. If F is an l-ideal in the Riesz space (E,C) and x 7→ [x] the quotient mapping from E to E/F , then

[C]F = { [x] ∈ E/F | there exists f ∈ F such that x + f ∈ C } determines a positive cone in E/F such that (E/F, [C]F ) is a Riesz space. The quotient mapping x 7→ [x] is an l-homomorphism. Remark. If F is just a Riesz subspace of (E,C), the quotient (E/F, C/F = [C]) is not always a Riesz space. The subspace F has to be order-convex in order that the cone [C] be proper. In other words, the sequence

q 0 → (F,C ∩ F ) →i (E,C) → (E/F, [C]) → 0 is algebraically exact for each Riesz subspace F , but it is exact in the sense of order (i.e. i and q are l-homomorphisms) only if F is an l-ideal. Q If { (Eα,Cα) | α ∈ A } is a family of RieszQ spaces, then the product α Eα is La Riesz space with respect to the cone C = α Cα and the algebraicQ direct sum α Eα is an l-ideal in (E,CL ). Moreover, each projection pα : α Eα → Eα (resp. each injection jα : Eα → α Eα ) is an l-homomorphism (resp. an l-isomorphism). A subset A in a Riesz space (E,C) is order-complete (resp. σ -order complete) if each net directed upwards (resp. increasing sequence) in A which is majorized in E has the supremum which belongs to A. In particular, for A = E , we say that (E,C) is (σ -) order-complete. Order-complete l-ideals in a Riesz space are called normal subspaces (bands). The following propositions are of particular interest in the algebraic theory of Riesz spaces. We state them without proofs.

Proposition 1.5. For an arbitrary subset A of a Riesz space (E,C), the set Ad is a normal subspace of E .

Proposition 1.6. Let (E,C) is an order-complete Riesz space and let A be a subset of E . Then E is the direct sum of normal subspaces Ad and Add , i.e. E = Ad ⊕ Add .

Proposition 1.7. For a Riesz space (E,C) the following conditions are equivalent: (a) (E,C) is Archimedean; (b) For each normal subspace A in E , A = Add ; 3.1. Basics of the theory of Riesz spaces 77

(c) For each l-ideal A in E , Add is the smallest normal subspace which contains A. Let now (E,C) be just an ordered vector space with the algebraic dual E∗ . A linear form f on E is order-bounded if it is bounded on order-intervals. It is positive if f(x) > 0 for all x ∈ C . Let Eb be the set of all order-bounded linear forms and C∗ the set of all positive linear forms. Then C∗ ⊂ Eb ⊂ E∗ and C∗ is a cone in E∗ ; Eb is a vector subspace of E∗ . The subspaces (C∗ − C∗,C∗) and (Eb,C∗) are known as the order-dual and order-bounded dual of the ordered vector space (E,C). Let us remark that the inclusion C∗ − C∗ ⊂ Eb can be strict ([73], p. 33). The next theorem is the one of important ones in the duality theory of ordered vector spaces: Theorem 1.8. (F. Riesz) Let (E,C) be a weak Riesz space. Then (Eb,C∗) is an order-complete Riesz space. Proof. Since (Eb,C∗) is a partially ordered vector space, it is suﬃcient to prove that for an arbitrary f ∈ Eb there exists sup(f, 0). If f ∈ Eb , deﬁne a mapping g : C → R with

g(u) = sup{ f(x) | x ∈ [0, u] }.

If u, v ∈ C we have

g(u) + g(v) = sup{ f(x) | x ∈ [0, u] } + sup{ f(y) | y ∈ [0, v] } = sup{ f(x) + f(y) | x ∈ [0, u], y ∈ [0, v] } = sup{ f(x + y) | x ∈ [0, u], y ∈ [0, v] } = sup{ f(z) | z ∈ [0, u + v] } = g(u + v), since in each Riesz space [0, u] + [0, v] = [0, u + v] holds. Thus, g is an additive function on the generating cone C . g is obviously also positive and homogeneous. If x ∈ E , then x = u−v for some u, v ∈ C . Then we can take g1(x) = g(u)−g(v). It is easy to check that g1 is a linear form on E and that it agrees with g on C . ∗ b b ∗ Evidently, 0, f 6 g1 , i.e. g1 ∈ C ⊂ E and g1 = sup{0, f}. Thus, (E ,C ) is a Riesz space and Eb = C∗ − C∗ . b ∗ We have to prove the order-completeness of the space (E ,C ). Let { fτ | τ ∈ D } is a majorized directed upwards net in Eb . For the given u ∈ C deﬁne ˜ ˜ h(u) = sup{ fτ (u) | τ ∈ D }. Then h is positive, homogeneous and additive on ˜ C because the net fτ is directed. Since C is generating, h has the unique linear b b extension h to E . Since the net fτ is majorized in E , it follows that h ∈ E . b Obviously, fτ 6 h for all τ ∈ D. On the other hand, if g ∈ E and fτ 6 g for all τ ∈ D, then h(u) = sup{ fτ (u) | τ ∈ D } 6 g(u) for all u ∈ C and so b ∗ h = sup{ fτ | τ ∈ D }. This means that (E ,C ) is an order-complete Riesz space. Corollary. If (E,C) is a weak Riesz space (in particular, a Riesz space) and f ∈ Eb , then the following assertions hold: 78 III. Ordered topological vector spaces

1◦ (∀u ∈ C)(∀f ∈ Eb) f +(u) = sup{ f(x) | 0 6 x 6 u }; 2◦ (∀u ∈ C)(∀f ∈ Eb) f −(u) = sup{ f(x) | −u 6 x 6 0 }; 3◦ (∀u ∈ C)(∀f ∈ Eb) |f|(u) = sup{ f(x) | −u 6 x 6 u } = sup{ |f(x)| | −u 6 x 6 u }; 4◦ (∀x ∈ E)(∀f ∈ Eb) |f(x)| 6 |f|(|x|); 5◦ (∀x ∈ E)(∀f ∈ Eb) sup(f, g)(u) = sup{ f(x) + g(u − x) | 0 6 x 6 u } = sup{ f(v) + q(w) | v, w > 0, u = v + w }; 6◦ (∀x ∈ E)(∀f ∈ Eb) inf(f, g)(u) = inf{ f(v) + q(w) | v, w > 0, u = v + w }; 7◦ (∀u ∈ C)[−u, u]◦ = { f ∈ Eb | |f|(u) 6 1 }, where [−u, u]◦ is the polar of [−u, u] taken in the space Eb .

Before stating the result dual to the one given in the previous corollary, the following lemma is of interest.

Lemma 1.9. Let (E,C) be a Riesz space and let f ∈ C∗ . Then for an arbitrary u ∈ C there exists g ∈ C∗ such that 0 6 g 6 f , g(u) = f(u) and g(x) = 0 whenever x ⊥ u. Proof. See [137].

Corollary 1. Let (E,C) be a Riesz space. Then for arbitrary f ∈ C∗ and x ∈ E the following holds: 1◦ f(x+) = sup{ g(x) | 0 6 g 6 f }; 2◦ f(x−) = sup{ h(x) | −f 6 h 6 0 }; 3◦ f(|x|) = sup{ g(x) | −f 6 g 6 f } = sup{ |g(x)| | −f 6 g 6 f }. Proof. It is suﬃcient to prove 1◦ since 2◦ and 3◦ follow directly from 1◦ . For an arbitrary g ∈ Eb satisfying 0 6 g 6 f it is g(x) 6 g(x+) 6 f(x+), hence sup{ g(x) | 0 6 g 6 f } 6 f(x+). The previous lemma implies that there exists h ∈ C∗ with 0 6 h 6 f , h(x+) = f(x+) and h(y) = 0 whenever y ⊥ x+ . Therefore f(x+) = h(x+) = h(x+) − h(x−) = h(x) and so f(x+) = sup{ g(x) | 0 6 g 6 f } follows.

Corollary 2. Let (E,C) be a Riesz space and Y a Riesz subspace of (Eb,C∗). Then the following holds: 1◦ If A is a solid subset of E , then its polar A◦ in Y is a solid subset of Y ; 2◦ If B is a solid subset of Eb , then its polar B◦ in E is a solid subset of E . Proof. Let g ∈ A◦ , f ∈ Y and |f| 6 |g|. If a ∈ A, then (using the Corollary of the Theorem 1.8) we have

f(a) 6 |f|(|a|) 6 |g|(|a|) = sup{ g(x) | |x| 6 |a| }.

Since A is a solid subset of E and since g ∈ A◦ , we have |g|(|a|) 6 1, i.e. f(a) 6 1. Thus, on the base of the Proposition 1.3, A◦ is a solid subset of Y . This proves 1◦ , and 2◦ can be proved in a similar way using the previous corollary. 3.2. Topological vector Riesz spaces 79

We say that a seminorm p on an ordered vector space (E,C) is a Riesz seminorm if it satisﬁes the following two conditions: 1◦ −y 6 x 6 y =⇒ p(x) 6 p(y) (absolute monotonicity); 2◦ for all x ∈ E satisfying p(x) < 1, there exists y ∈ C with p(y) < 1, such that −y 6 x 6 y . It is not hard to prove the following Proposition 1.10. For each Riesz space (E,C) and a seminorm p on it, the following conditions are equivalent: (a) p is a Riesz seminorm; (b) p is monotonic and p(|x|) = p(x) for all x ∈ E ; (c) from |x| 6 |y| it follows p(x) 6 p(y). The condition (c) is often used as a deﬁnition of a Riesz seminorm.

3.2. TOPOLOGICAL VECTOR RIESZ SPACES

Definition 1. By a topological vector Riesz space, abbreviated tvRs, (or a topological vector lattice) we shall understand a Riesz space (E,C) endowed with a Hausdorﬀ vector topology t which has a base of neighbourhoods of the origin formed by solid subsets. If, moreover, (E, t) is an lcs, then we shall say that the triple (E, C, t) is a locally convex Riesz space (abbreviated lcRs). The following propositions describe topological vector Riesz spaces in more details. Proposition 2.1. For each Riesz space (E,C) endowed with a Hausdorﬀ vector topology t the following conditions are equivalent: (a) (E, C, t) is a topological vector Riesz space. (b) The mapping (x, y) 7→ sup(x, y) is uniformly continuous on E × E . (c) The mapping x 7→ x+ = sup(x, 0) is uniformly continuous on E .

(d) For arbitrary two nets { xα | α ∈ D } and { yα | α ∈ D } in E satisfying |xα| 6 |yα| for all α ∈ D, if yα →t 0, then xα →t 0. Proof. See [103] or [137]. Proposition 2.2. Let (E, C, t) be a tvRs. Then the following holds: 1◦ C is a t-closed subset. 2◦ (E,C) is an Archimedean Riesz space; 3◦ The solid cover of each t-bounded subset is t-bounded, i.e. if A is t- bounded, such are |A|, A+ and A− , where

|A| = { |a| | a ∈ A },A+ = { a+ | a ∈ A },A− = { a− | a ∈ A }. 80 III. Ordered topological vector spaces

4◦ C is a strict B -cone in (E, t), i.e. each t-bounded subset A is contained in the set of the form B ∩ C − B ∩ C for a certain t-bounded subset B . 5◦ The t-closure of each Riesz subspace (resp. l-ideal) of E is a Riesz subspace (resp. l-ideal). 6◦ The t-closure of each solid subset is a solid subset. 7◦ The solid kernel sk(A) of each t-closed set A is t-closed. 8◦ A subset V in E is t-bornivorous if and only if sk(V ) is t-bornivorous (t-bornivorous means “absorbs all t-bounded subsets”). 9◦ Each normal subspace of E is t-closed. 10◦ If (E, C, t) is order-complete, then (E, t) is the direct topological sum of the subspaces (B, t|B) and (Bd, t|Bd) for each normal subspace B of E . Proof. We shall prove some of these easy properties. 1◦ Since C = { x ∈ E | x− = 0 } and the mapping x 7→ sup(−x, 0) = x− is continuous, it follows that C is a t-closed subset of E . ◦ 1 2 Let nx 6 y for all n ∈ N. We have to show that x 6 0. Since n y →t 0 1 t and n y − x ∈ C = C , then −x ∈ C , i.e. x 6 0. (Remark that the order in each ordered vector space (E, C, t) is Archimedean if the cone C is t-closed.) ◦ T 6 Since A = U∈U A+U , where U is a base of neighbourhoods of the origin, then A is a solid subset if A is such, because it is an intersection of solid subsets (Proposition 1.4.5 ◦ ). 9◦ Let F be a normal subspace of E . Then, by the Proposition 1.7 (since (E,C) is Archimedean), we have F = F dd . The closedness of each set of the form Ad is a consequence of the continuity of lattice operations. That is why F is a closed subspace. 10◦ follows from the Propositions 1.6 and 2.1. An lcRs (E, C, t) is a Banach lattice (resp. Fr´echetlattice) if (E, t) is a Banach (resp. Fr´echet) lcs. Similarly to the locally convex case, each lcRs (E, C, t) is l- isomorphic and topologically isomorphic to a Riesz subspace of a projective limit of Banach lattices ([137], p. 140). Besides, each Riesz subspace F of an lcRs (E, C, t) is an lcRs in the induced topology (according to (c) of the Proposition 2.1). Similarly, the class of lcRs’s is stable with respect to arbitrary products, direct sums and quotients by closed l-ideals. A tvRs (E, C, t) is locally order-complete if there exists a base of t- neighbourhoods of the origin formed by solid and order-complete sets. Evidently, each locally order-complete Riesz space is order-complete. We say that (E, C, t) is boundedly order-complete if each directed net in E which is t-bounded has the supremum in E . Concerning the introduced notions, we state two deeper properties from the theory of Riesz spaces.

Proposition 2.3. (H. Nakano) Let (E, C, t) be a tvRs. If (E, C, t) is locally order-complete, then each order-interval in E is t-complete. 3.2. Topological vector Riesz spaces 81

Theorem 2.4. (H. Nakano) A tvRs (E, C, t) is t-complete if it is locally and boundedly order-complete. That the converse of the preceding theorem does not hold is shown by the following

Example. Let c0 be the space of real zero-sequences with the usual norm and ordered by coordinates. Then c0 is a Banach lattice. Let en = (1, 1,..., 1, 0, 0,... ) ∈ c and A = { e | n ∈ N }. Then A is directed and norm- | {z } 0 n n bounded, but it obviously has no supremum in c0 . Therefore, c0 is not boundedly order-complete. Remark. The topology of each topological vector (resp. locally convex) Riesz space is determined by a family { pα | α ∈ D } of continuous Riesz F -seminorms (resp. seminorms). Namely, each solid neighbourhood of the origin V generates a solid topological string (Vn)n∈N , V1 = V , where the F -seminorm associated to (Vn) is constructed as in the Theorem I.2.4. In the locally convex case, the Minkowski functional of each convex and solid neighbourhood of the origin is a continuous Riesz seminorm. We finish this paragraph with a few examples. ◦ Examples. 1 The family of all strings (Vn)n∈N , where each knot is a solid subset of a Riesz space (E,C), generates the finest vector topology t such that (E, C, t) is a tvRs. 2◦ The set of all convex, solid and absorbing subsets of a Riesz space (E,C) forms a base of neighbourhoods of the origin of the finest locally convex topology t on E such that (E, C, t) is an lcRs. 3◦ If (E, C, t) is an lcRs with the topological dual E0 and the set of all positive t-continuous linear forms C0 , then there exist locally convex topologies on E and E0 , respectively, of uniform convergence on the families of order-bounded 0 0 0 subsets of Riesz spaces (E ,C ) and (E,C). They are denoted by σs(E,E ) and 0 0 σs(E ,E), respectively, and called Dieudonnétopologies;(E, C, σs(E,E )) and 0 0 0 0 0 (E ,C , σs(E ,E)) are lcRs’s. Topology σs(E,E ) (resp. σs(E ,E)) can also be characterized as the coarsest solid topology on E (resp. E0 ) which is finer than the weak topology σ(E,E0) (resp. σ(E0,E)). 4◦ If (E, C, t) is an lcRs, then (E0,C0, β(E0,E)) is an lcRs. 5◦ The completion (E,˜ C,˜ t˜) of an lcRs (E, C, t) is an lcRs, where the cone C˜ is the closure of C in the completion (E,˜ t˜). The example 4◦ is particularly important in the duality theory of lcRs’s. Namely, if (E, C, t) is an arbitrary lcRs, then its strong dual (E0,C0, β(E0,E)) is an order-complete lcRs and E0 is an l-ideal in Eb . Moreover, it is also locally order-complete. If (E, C, t) is a quasibarrelled lcRs, then (E0,C0, β(E0,E)) is boundedly order-complete and then, by the Nakano’s theorem, the strong dual is complete. 82 III. Ordered topological vector spaces

From the previous observations it follows that, to the contrary of the case of locally convex spaces (Y. K¯omura [61]), in the class of reﬂexive lcRs’s there are no incomplete spaces. Thus, if (E, C, t) is an lcRs, then we have two sequences of dual spaces:

E ⊂ E00 ⊂ E(IV ) ⊂ ... ⊂ E0b ⊂ E0∗ and E0 ⊂ E000 ⊂ E(V ) ⊂ ... ⊂ Eb ⊂ E∗. All these spaces (except possibly E ) are order-complete l-ideals in E0b , resp. Eb (Theorem 1.8). E is an order-complete l-ideal in E00 if and only if the Dieudonn´e 0 0 topology σs(E ,E) is compatible with respect to the dual pair hE,E i. That E is a Riesz subspace of (E00,C00) follows from Proposition 2.5. Let (E, C, t) be an lcRs with the strong dual (E0,C0, β(E0,E)) and the bidual (E00,C00). Then the image Eˆ of E in E00 with respect to the embedding x 7→ xˆ given by

xˆ(f) = f(x) for each f ∈ E0 , is a Riesz subspace of E00 and the given mapping is an l-isomorphism from E onto Eˆ . Proof. The given mapping is obviously a bijection from E onto Eˆ . Let us show that xc+ = (ˆx)+ . Let f ∈ C0 . The Corollary 1 of Lemma 1.9 implies that

xc+(f) = f(x+) = sup{ g(x) | 0 6 g 6 f }.

Obviously,x ˆ is an order-bounded linear form on E0 . Then by the Corollary of the Theorem 1.8 we obtain

(ˆx)+(f) = sup{ xˆ(g) | 0 6 g 6 f }.

Therefore, xc+(f) = (ˆx)+(f), and so xc+ = (ˆx)+ , since E0 = C0 − C0 . It is proved that the image Eˆ of E in E00 is a Riesz subspace.

3.3. THE BASIC CLASSES OF LOCALLY CONVEX RIESZ SPACES

If (E, C, t) is an lcRs, then it is called a barrelled (resp. bornological, quasibarrelled, ultrabornologique) locally convex Riesz spaces if (E, t) is of the corresponding type as a locally convex space. Each of the classes of such spaces is stable with respect to the quotient by a closed l-ideal. Concerning the stability with respect to l-ideals, we state ﬁrst of all the following 3.3. The basic classes of locally convex Riesz spaces 83

Proposition 3.1. For each lcRs (E, C, t) the following conditions are equivalent: (a) (E, C, t) is a bornological lcRs; (b) Each convex and solid set in E which absorbs t-bounded subsets is a t- neighbourhood of the origin; (c) Each topologically bounded Riesz seminorm p on E is t-continuous. Proof. The implication (a) =⇒ (b) is trivial. Since a subset V of E absorbs all t-bounded subsets if and only if sk(V ) absorbs all t-bounded subsets, we obtain that (b) =⇒ (a). The implication (b) =⇒ (c) follows from the fact that the set V = { x ∈ E | p(x) 6 1 } is a convex solid subset of E which absorbs all t-bounded subsets. Finally, since the Minkowski functional of each convex solid and absorbing subset of E is a Riesz seminorm, we conclude that (c) =⇒ (b), which completes the proof. Using the characterization of bornological lcRs’s given in (b) of the previous Proposition, we can prove

Theorem 3.2. (I. Kawai [57]) The class of bornological lcRs’s is stable with respect to each l-ideal. Proof. Let F be an l-ideal in a bornological lcRs (E, C, t) and let V be a convex solid subset of F which absorbs all t|F -bounded subsets. It is suﬃcient to prove that V is a t|F -neighbourhood of the origin. Consider the subset U := { x ∈ E | [0, |x|] ∩ F ⊂ V } of E . Then the set U is convex, solid and U ∩ F = V . Let us prove that U absorbs all t-bounded subsets. If this is not the case, then there exists a solid t-bounded subset B , such that B 6⊂ nU for all n ∈ N. This means that there exist bn ∈ B with bn ∈/ nU , n ∈ N. Hence, there exist yn ∈ F such that 0 6 yn 6 |bn/n| and yn ∈/ V , n ∈ N. Since B is a solid set, the set { nyn | n ∈ N } is contained in B ∩ F , which contradicts the fact that V abosrbs B ∩ F . We have thus proved that U is t-bornivorous in E , i.e. V = U ∩ F is a t|F -neighbourhood of the origin, which means that the l-ideal F is a bornological lcRs in the induced topology. We say that an lcs (E, t) is boundedly closed if each linear form on E is t-continuous provided it is bounded on t-bounded subsets (E0 = E˜0 ). Each bornological lcs possesses this property and each locally bounded Mackey space is bornological. Recall that for an arbitrary lcs (E, t),(E, τ(E, E˜0)) is the associated bornological space. If (E, C, t) is an lcRs, then (E, C, τ(E, E˜0)) is also an lcRs. Indeed, if U is a τ(E, E˜0)-neighbourhood of the origin , then it absorbs all t- bounded, closed, solid and convex subsets, which by the Proposition 2.2.8 ◦ means that sk(U) is t-bornivorous, hence a τ(E, E˜0)-neighbourhood of the origin. Thus, (E, C, τ(E, E˜0)) is an lcRs. These conclusions and the previous theorem lead us to

Proposition 3.3. The class of boundedly closed lcRs’s is stable with respect to arbitrary l-ideals. 84 III. Ordered topological vector spaces

Proof. Let F is an l-ideal in a boundedly closed lcRs (E, C, t). Since E0 = E˜0 , then, as it was just said, (E, C, τ(E, E˜0)) is an lcRs. On the basis of the previous Theorem, (F,C ∩ F, τ(E, E˜0)|F ) is a bornological lcRs. Therefore τ(E, E˜0)|F = τ(F,F 0) = τ(F, F˜0). Since σ(F,F 0) 6 t|F 6 τ(F,F 0) = τ(F, F˜0), we have F 0 = F˜0 , i.e. (F,C ∩ F, t|F ) is boundedly closed. A subset V of an lcRs (E, C, t) is called a solid barrel, if V is t-closed, convex, solid and absorbing. By the already mentioned Proposition 2.2.8 ◦ , we know that the barrel V absorbs all t-bounded subsets if and only if the solid barrel sk(V ) absorbs all t-bounded subsets. The following proposition can be proved similarly to the Proposition 3.1. Proposition 3.4. For each lcRs (E, C, t) the following conditions are equivalent: (a) (E, C, t) is a quasibarrelled lcRs; (b) Each solid barrel in (E, C, t) which absorbs t-bounded subsets is a t- neighbourhood of the origin; (c) Each topologically upper semibounded Riesz seminorm on E is t- continuous. Now, similarly to the Theorem 3.2, the following result can be proved. Theorem 3.5. (I. Kawai [57]) The class of quasibarrelled lcRs’s is stable with respect to arbitrary l-ideals. Proof. We have only to prove that the set U which was defined in the proof of the Theorem 3.2, is closed, i.e. that it is a bornivorous solid barrel in E , if V possessed such properties in F . Actually, if x ∈ U , then there exists a net { xτ | τ ∈ D } in U such that xτ →t x. Let y ∈ F be such that 0 6 y 6 |x|. Then 0 6 inf(y, |xτ |) 6 |xτ | ∈ U and each inf(y, |xτ |) ∈ F (since F is an l-ideal). By the definition of the set U it follows that inf(y, |xτ |) ∈ V . Using the continuity of lattice operations sup and inf we conclude that inf(y, |xτ |) → inf(y, |x|) = y , i.e. y ∈ V = V . This means that x ∈ U and the set U is closed. If the space (E, C, t) is quasibarrelled, U is then a t-neighbourhood of the origin and so V is a t|F -neighbourhood of the origin and (F, t|F ) is a quasibarrelled lcRs. The following example shows that the class of barrelled (resp. ultrabornologique) lcRs’s is not stable with respect to arbitrary l-ideals. Example. Let E be the Banach space of all continuous real-valued functions on [0, 1] with the sup-norm defined by kxk = max{ |x(t)| | t ∈ [0, 1] } and let C = { x ∈ E | x(t) > 0 for all t ∈ [0, 1] }. Then (E,C, k k) is a Banach lattice. Let, furthermore, F be the vector subspace of E containing all the functions x which are equal to zero in a neighbourhood of the point t = 0 (depending on x), and CF = C ∩ F . Then F is an l-ideal and (F,CF , k k) is a normed Riesz space which is not barrelled (see the Example 2◦ in paragraph 1.5). Actually, the set D = { x ∈ F | n|x(1/n)| 6 1 for all n ∈ N } is a solid barrel in (F,CF , k k), but it is not a neighbourhood of the origin in F , as was shown in the mentioned 3.3. The basic classes of locally convex Riesz spaces 85

example. We conclude that (F,CF , k k) is an lcRs which is not barrelled (nor ultrabornologique). In the class of locally convex Riesz spaces, in a way more natural than the quasibarrelled and bornological are the classes of order-quasibarrelled and order- bornological spaces. We say that an lcRs (E, C, t) is order-bornological (resp. order-quasibarrelled) if each absolutely convex set (resp. each barrel) in it which absorbs order-bounded subsets is a t-neighbourhood of the origin. Both classes are stable with respect to quotients by arbitrary l-ideals. Since each solid barrel absorbs all order-bounded subsets, the previous example shows that the classes of order-quasibarrelled and order-bornological spaces are not stable with respect to arbitrary l-ideals. The following table shows the relationships between the basic classes of locally convex Riesz spaces.

ultrabornologique   y barrelled   y order-bornological −−−−→ order-quasibarrelled     y y bornological −−−−→ quasibarrelled

We shall now consider the stability of the introduced classes with respect to l-ideals.

We say that an l-ideal F in an lcRs (E, C, t) is σ -normal if from un ∈ F , n ∈ N and 0 6 un ↑ u in E it follows that u ∈ F . Proposition 3.6. Let (E, C, t) be an order-quasibarrelled σ -order complete lcRs and F its σ -normal subspace. Then F is order-quasibarrelled with respect to the induced topology and order. Proof. Let V be a solid barrel in the subspace F and put

U = { x ∈ E | [0, |x|] ∩ F ⊂ V }.

Then U is a t-closed, convex and solid set in E such that U ∩ F = V (Theorems 3.2 and 3.5). If U is not absorbing in E , then there exists x ∈ C such that for all 1 n ∈ N there exists yn ∈ F for which 0 6 yn 6 n x, but yn ∈/ V . Using the σ -order completeness of (E, C, t), we conclude that there exists y = sup{ nyn | n ∈ N } in E . The σ -normality of the subspace F implies that y ∈ F . Now { nyn | n ∈ N } is contained in the order-interval [0, y] in F but it is not absorbed by V . But this is impossible since V is a solid barrel in F (it absorbs order-bounded subsets). This means that the extension U of the set V from F to E is a solid barrel in E , 86 III. Ordered topological vector spaces i.e. a t-neighbourhood of the origin. Since U ∩ F = V , V is a t|F -neighbourhood of the origin, i.e. F is an order-quasibarrelled space in the induced topology and order. The following proposition can be proved in a similar way.

Proposition 3.7. The class of order-bornological (resp. barrelled) σ -order- complete lcRs’s is stable with respect to arbitrary σ -normal l-ideals.

Besides order-bornological and order-quasibarrelled spaces some other classes of barrelled-type spaces can be introduced within locally convex Riesz spaces. Du- ally speaking, these spaces are characterized by the relationship of equicontinuous 0 subsets in the dual with some kind of σs(E ,E)-bounded subsets. An lcRs (E, C, t) is called order-countably quasibarrelled (resp. order-σ - 0 quasibarrelled; order-sequentially quasibarrelled) if each σs(E ,E)-bounded subset in E0 is equicontinuous, provided it is a countable union of t-equicontinuous subsets 0 0 (resp. it is a σs(E ,E)-bounded sequence; σs(E ,E)-zero sequence). Neither of these classes is stable when passing to an arbitrary l-ideal. An lcRs (E, C, t) is said to be countably barrelled (resp. σ -quasibarrelled; sequentially quasibarrelled) if the lcs (E, t) is of the corresponding type. Before proving the hereditary properties of the introduced classes, we shall state some auxiliary assertions.

Proposition 3.8. [84] For an arbitrary lcRs (E, C, t) the following conditions are equivalent: (a) (E, C, t) is countably quasibarrelled. (b) Each solid d-barrel which absorbs all t-bounded subsets of E is a t- neighbourhood of the origin. (c) Each solid d-barrel which absorbs all positive t-bounded subsets of E is a t-neighbourhood of the origin. (d) Each β(E0,E)-bounded subset of E0 which is a countable union of t- equicontinuous subsets of E0 is t-equicontinuous. (e) Each positive β(E0,E)-bounded subset of E0 which is a countable union of t-equicontinuous subsets of E0 is t-equicontinuous. Proof. (d) ⇐⇒ (e) because (E0,C0, β(E0,E)) is an lcRs. Further, (a) ⇐⇒ (d) using the dual characterization of countably quasibarrelled spaces, see the begining ◦ of the paragraph 2.2. (b) ⇐⇒ (c) on the base of the PropositionT 2.2.4 and (a) =⇒ (b) is trivial. It remains to prove (b) =⇒ (a).T If V = n∈N Vn is a bornivorous d-barrel in the space (E, C, t), then sk(V ) = n∈N sk(Vn) is a solid bornivorous d-barrel in (E, C, t) and so a t-neighbourhood of the origin. Hence, V is a t-neighbourhood of the origin, too. Let us remark that (sk(Vn)) is a sequence of t-neighbourhoods of the origin (closed and absolutely convex) because of the fact that in an lcRs a subset U is a t-neighbourhood of the origin if and only if sk(U) is a t-neighbourhood of the origin. The proof is complete. 3.3. The basic classes of locally convex Riesz spaces 87

By a solid σ -barrel in an lcRs (E, C, t) we shall mean a barrel of the form T 0 n∈N Vn , where (Vn) is a sequence of σs(E,E )-closed convex solid neighbourhoods of the origin.

Proposition 3.9. [84] For each lcRs (E, C, t) the following conditions are equivalent: (a) (E, C, t) is a σ -quasibarrelled lcRs. (b) Each solid σ -barrel which absorbs all t-bounded subsets in E is a t- neighbourhood of the origin. (c) Each β(E0,E)-bounded sequence in E0 is t-equicontinuous. (d) Each positive β(E0,E)-bounded sequence in E0 is t-equicontinuous. Proof. As far as (E0,C0, β(E0,E)) is an lcRs, (c) ⇐⇒ (d) holds on the base of the Proposition 2.2.3 ◦ . The implication (a) =⇒ (b) and equivalence (a) ⇐⇒ (c) T∞ are trivial, so it remains to prove (b) =⇒ (a). Let V = n=1 Vn be a bornivorous σ -barrel in the space (E, C, t). On the base of the Proposition 2.2.8 ◦ , sk(V ) and sk(Vn) are also bornivorous subsets of E . Moreover,

\∞ \∞ 0 0 ◦ sk(V ) = sk(Vn) = [−xn, xn] n=1 n=1

0 0 holds for a certain sequence xn ∈ E , i.e. sk(V ) is a bornivorous solid σ -barrel, and so a t-neighbourhood of the origin. Since sk(V ) ⊂ V , V is also a t-neighbourhood 0 0 ◦ of the origin and the proof is complete. That sk(Vn) = [−xn, xn] follows from [137], pp. 155–156. Before proving the propositions about l-ideals of countably quasibarrelled, σ - quasibarrelled and sequentially quasibarrelled lcRs’s, we give the following auxiliary assertion.

Proposition 3.10. Let F be an l-ideal in an lcRs (E, C, t) and V a convex solid neighbourhood of the origin in F . Then there exists a t-neighbourhood of the origin U such that U ∩ F = V . Proof. Put U := { x ∈ E | [0, |x|] ∩ F ⊂ V }. The set U is convex, solid and U ∩F = V . We have only to prove that U is a t-neighbourhood of the origin. If it is not the case, then W 6⊂ U for each convex solid t-neighbourhood of the origin W . For each W ∈ U , where U denotes the family of all convex solid t-neighbourhoods of the origin, there exists xW ∈ W such that xW ∈/ U . Hence there exist yW ∈ F , 0 6 yW 6 |x| such that yW ∈/ V . Notice that { xW | W ∈ U, ⊃ } is a net in (E, C, t) which tends to 0. Since W is a solid subset, it follows that yW ∈ W and the net { yW | W ∈ U, ⊃ } tends also to 0 in (E, C, t) but it does not in (F, t|F ), contrary to the fact that yW ∈ W ∩ F for all W ∈ U .

Theorem 3.11. [84] An arbitrary l-ideal of a (DF) (resp. countably quasibarrelled) lcRs is a space of the same type in the induced topology. 88 III. Ordered topological vector spaces

T∞ Proof. Let F be an l-ideal in an lcRs (E, C, t) and let V = n=1 Vn be a bornivorous solid d-barrel in (F,CF , t|F ). Let, in addition,

Un := { x ∈ E | [0, |x|] ∩ F ⊂ Vn },U := { x ∈ E | [0, |x|] ∩ F ⊂ V }.

T∞ It is easy to check, using the previous Proposition, that U = n=1 Un is a bornivorous solid d-barrel in E and that U ∩ F = V . Of course, if (E, C, t) has a fundamental sequence of t-bounded subsets, then (F,CF , t|F ) has the same property. This completes the proof. 0 Let us observe that Dieudonnétopology σs(E,E ) of an lcRs (E, C, t) induces 0 Dieudonnétopology σs(F,F ) on each l-ideal (F,CF , t|F ). Indeed, since the image of each order-bounded subset of (E0,C0) by the quotient mapping q : E0 → E0/F ◦ 0 0 0 is order-bounded, it is obvious that σs(E,E )|F 6 σs(F,F ). But σs(F,F ) is the coarsest solid topology which is finer than σ(F,F 0) = σ(E,E0)|F , and so 0 0 σs(E,E )|F = σs(F,F ). Using this observation, we obtain the following result: Theorem 3.12. [84] The class of σ -quasibarrelled lcRs’s is stable with respect to arbitrary l-ideals. Proposition 3.13. For each lcRs (E, C, t) the following conditions are equivalent: (a) (E, C, t) is sequentially quasibarrelled. (b) Each positive β(E0,E)-zero sequence in E0 is t-equicontinuous. 0 (c) For each sequence (Un) of closed absolutely convex σ(E,E )-neighbourhoodsT of the origin, such that each t-bounded subset in E is contained in U for T n>m n some m, the set n>1 Un is a t-neighbourhood of the origin. 0 (d) For each sequence (Vn) of closed solid convex σs(E,E )-neighbourhoodsT of the origin, such that each t-bounded subset in E is contained in V for T n>m n some m, the set n>1 Vn is a t-neighbourhood of the origin. 0 0 0 0 Proof. (b) =⇒ (a). Since (E ,C , β(E ,E)) is an lcRs, a sequence (xn) in E 0 0 is a β(E ,E)-zero sequence if and only if (|xn|) is a β(E ,E)-zero sequence, i.e. if + − 0 0 and only if (xn ) and (xn ) are β(E ,E)-zero sequences. So, if (xn) is a β(E ,E)- + − + − zero sequence, the implication follows because xn = xn − xn and |xn| = xn + xn . 0 (a) =⇒ (c). If (Un) is a sequence of closed absolutely convex σ(E,E )- neighbourhoods of the origin, then U = A◦ , where A = {f n, . . . , f n } is a finite n n n 1 kn 0 T subset of E . Let A ⊂ n>m Un for some m = m(A). Now we have [ [ [ ◦ ◦ ◦◦ A ⊃ Un = An ⊃ An. n>m n>m n>m

It follows that the sequence

1 1 2 2 n n (f1 , . . . , fk1 ; f1 , . . . , fk2 ; ... ; f1 , . . . , fkn ; ... ) 3.3. The basic classes of locally convex Riesz spaces 89

S of elements of the set A is a β(E0,E)-zero sequence. From (a) we have ¡ ¢ n>1 n S ◦ T ◦ T that n>1 An = n>1 An is a t-neighbourhood of the origin, i.e. n>1 Un is a T T ◦ t-neighbourhood of the origin, since n>1 Un = n>1 An holds. This proves that (a) =⇒ (c). 0 (c) =⇒ (d). Let (Vn) be a sequence of closed solid convex σs(E,E )- neighbourhoods of theT origin which satisfy the condition: each t-bounded subset A of E is contained in n>m Vn for some m = m(A). From [137], pp. 155–156 we ◦ know that Vn = [−xn, xn] = sk(Un), where Un = { x ∈ E | |xn(x)| 6 1 }. Since \ \ µ \ ¶ \ A ⊂ Vn = sk(Un) = sk Un ⊂ Un, n>m n>m n>m n>m T it follows that the sequence (U ) satisﬁes (c); hence, U is a t-neighbourhood ¡T n ¢ T Tn>1 n of the origin. Thus, sk n>1 Un = n>1 sk(Un) = n>1 Vn is a t-neighbourhood of the origin, because in each lcRs, W is a neighbourhood of the origin if and only if sk(W ) is a neighbourhood of the origin . This proves the implication (c) =⇒ (d). 0 0 0 (d) =⇒ (b). Let (xn) be a β(E ,E)-zero sequence, such that xn ∈ C ⊂ E . If ◦ A is a solid convex t-bounded subset of E , then it is xn , −xn ∈ A for n > m(A), ◦ ◦ ◦◦ i.e. [−xn, xn] ⊂ A . Then it follows that [−xn, xn] ⊃ A ⊃ A for n > m(A), T ◦ ◦ i.e. n>m[−xn, xn] ⊃ A. Since ([−xn, xn] ) is a sequence of closed solid convex 0 T ◦ σs(E,E )-neighbourhoods of the origin, from (d) it follows that n>1[−xn, xn] is a t-neighbourhood of the origin. From [ [ [ µ \ ¶◦ ◦◦ ◦ {xn} ⊂ [−xn, xn] ⊂ [−xn, xn] ⊂ [−xn, xn] n>1 n>1 n>1 n>1

0 we have that the sequence (xn) is a t-equicontinuous subset of E . This completes the proof of the proposition. Now we can prove the theorem about l-ideals in sequentially quasibarrelled lcRs’s. Theorem 3.14. [86] An arbitrary l-ideal in a sequentially quasibarrelled lcRs is of the same type with respect to the induced topology.

Proof. Let F be an l-ideal in an lcRs (E, C, t) and let (Vn) be a sequence 0 of closed solid convex σs(F,F )-neighbourhoodsT of the origin such that each t|F - bounded subset of F is contained in V for some m. On the base of (d) of the n>m n T previous proposition, it is suﬃcient to prove that n>1 Vn is a t|F -neighbourhood of the origin. Let

Un = { x ∈ E | [0, |x|] ∩ F ⊂ Vn }, n = 1, 2,... and U = { x ∈ E | [0, |x|] ∩ F ⊂ V }. T It can be easily seen that U = n>1 Un . The proof that for each n ∈ N Un is 0 a closed solid and convex σs(E,E )-neighbourhood of the origin is the same as in 90 III. Ordered topological vector spaces the propositions on bornological and quasibarrelled spaces. Using again (d) of the previousT proposition, we have to prove that each t-bounded subset is contained in n>m Un for some m. If this is not the case, then there exists a t-bounded m m T subset A such that A 6⊂ U for m = 1, 2,... , where U = n>m Un . It follows m that there exists a sequence (am) of elements from A such that am ∈/ U . Since m m m T U ∩ F = V , V = n>m Vn , there exist bm ∈ F , m ∈ N such that 0 6 m bm 6 |am| and bm ∈/ V . But, bm ∈ A ∩ F , because A is a solid subspace of E . m Thus, we have A ∩ F 6⊂ V for m = 1, 2,...T, which contradicts the fact that each t|F -bounded subset of F is contained in V for some m. It means that T n>m Tn ¡T ¢ n>1 Un is a t-neighbourhood of the origin, i.e. n>1 Vn = F ∩ n>1 Un is a t|F -neighbourhood of the origin, and (F,C ∩F, t|F ) is a sequentially quasibarrelled lcRs.

3.4. L-IDEALS OF TOPOLOGICAL VECTOR RIESZ SPACES

Let (E, C, t) be a tvRs in the sense of the Proposition 2.1. We say that it is an ultrabarrelled (resp. quasiultrabarrelled, ultrabornological) Riesz space if the topological vector space (E, t) is such in the sense of deﬁnitions from paragraph 2.3. Similarly the other classes of tvRs’s (ultra-(DF), ultra-Db , locally topological Riesz spaces, ... ) are introduced. A string (Vn)n∈N in a tvRs (E, C, t) is said to be solid if all of its knots Vn are solid. It is order-bornivorous if each knot Vn absorbs all order-bounded subsets. If each knot of the string is solid and closed, it is called a solid ultrabarrel. If (E,C) is a Riesz space, then all solid strings in (E,C) generate the ﬁnest linear solid topology T on E , such that (E,C,T ) is a tvRs. Similarly, all order- bornivorous strings in a Riesz space (E,C) generate the linear solid topology tb such that (E, C, tb) is a tvRs. We say that a tvRs (E, C, t) is order-ultrabornological if each order-bornivorous string in E is t-topological, i.e. if t = tb holds. A tvRs (E, C, t) is order-quasiultrabarrelled if each order-bornivorous ultrabarrel in it is t-topological. Using the Example given after Theorem 3.5, it can be shown that the classes of ultrabarrelled, order-ultrabornological and order-quasiultrabarrelled tvRs’s are not stable with respect to arbitrary l-ideals. It is easy to check that they are stable with respect to arbitrary quotients by closed l-ideals. Similarly as for lcRs’s, for ultrabornological and quasiultrabarrelled tvRs’s the following holds: Theorem 4.1. (D. Keim [58]) The class of ultrabornological (resp. quasiultrabarrelled) tvRs’s is stable with respect to arbitrary l-ideals. Before proving this Theorem, we state the following Proposition 4.2. A tvRs (E, C, t) is ultrabornological (resp. quasiultrabarrelled) if and only if each solid bornivorous string (resp. solid bornivorous ultrabarrel) in it is t-topological. 3.4. l -ideals of topological vector Riesz spaces 91

Proof. Let (Vn)n∈N be a bornivorous string (resp. bornivorous ultrabarrel) in (E, C, t). It is suﬃcient to prove that (sk(Vn))n∈N is a solid bornivorous string (resp. solid bornivorous ultrabarrel). The subset sk(Vn) is bornivorous (resp. closed) by the Proposition 2.2.8 ◦ and 7◦ . It remains only to prove that sk(Vn+1) + sk(Vn+1) ⊂ sk(Vn) for n = 1, 2,... . So let z ∈ sk(Vn+1) + sk(Vn+1). Then there are x, y ∈ sk(Vn+1) such that z = x + y . By the deﬁnition of the solid kernel it follows that there exist u, v ∈ C such that x ∈ [−u, u] ⊂ Vn+1 , y ∈ [−v, v] ⊂ Vn+1 , wherefrom we obtain that

x + y ∈ [−(u + v), u + v] ⊂ Vn+1 + Vn+1 ⊂ Vn, i.e. x ∈ sk(Vn). Proof of the Theorem. Let (F,C ∩F, t|F ) be an l-ideal of an ultrabornological (resp. quasiultrabarrelled) tvRs (E, C, t) and let V = (Vn)n∈N be a solid bornivorous string (resp. solid bornivorous ultrabarrel) in it. By the previous Proposition, it suﬃces to prove that the string V is t|F -topological. For each n ∈ N deﬁne the subset Un of E in the folowing way:

Un := { x ∈ E | [0, |x|] ∩ F ⊂ Vn }.

The subsets Un are solid, bornivorous (resp. closed) as it was shown in Theorems 3.2 and 3.5. It remains to prove that U = (Un) is a string in the space E , wherefrom it will follow that U is a solid bornivorous string (resp. solid bornivorous ultrabarrel) with the property U ∩ F = V . In other words, we have to prove that Un+1 + Un+1 ⊂ Un for each n ∈ N. Let x = a + b, where a, b ∈ Un+1 and y ∈ F such that 0 6 y 6 |x| = |a + b| 6 |a| + |b|. Since [0, |a| + |b|] = [0, |a|] + [0, |b|] holds, we have that y = y1 + y2 ∈ Vn+1 + Vn+1 ⊂ Vn , i.e. x ∈ Un . For other classes of tvRs’s we state the following results: Proposition 4.3. Let (E, C, t) be a tvRs and consider the following statements: 1◦ (E, C, t) is countably quasiultrabarrelled. 2◦ Each solid bornivorous σ -ultrabarrel in (E, C, t) is t-topological. 3◦ (E, C, t) is locally topological. 4◦ Each solid locally topological string in (E, C, t) is t-topological. 5◦ (E, C, t) is ultra-b-barrelled. 6◦ Each closed solid locally topological string in (E, C, t) is t-topological. Then 1◦ ⇐⇒ 2◦ , 3◦ ⇐⇒ 4◦ and 5◦ ⇐⇒ 6◦ . ¡ ¢ ◦ ◦ ◦ ◦ T (j) Proof. 1 =⇒ 2 is trivial; let us prove that 2 =⇒ 1 . Let n∈N Vn j∈N (j) is a bornivorous σ -ultrabarrel, where (Vn )j∈N is a closed topological string for (j) each j ∈ N. Then (sk(Vn ))j∈N is a closed solid topological string and using the equality µ \ ¶ \ (j) (j) sk Vn = sk(Vn ) n∈N n∈N 92 III. Ordered topological vector spaces

¡T (j) ¢ we obtain that n∈N sk(Vn ) j∈N is a solid bornivorous σ -ultrabarrel, which proves that 2◦ =⇒ 1◦ . The proofs for 3◦ ⇐⇒ 4◦ and 5◦ ⇐⇒ 6◦ are similar. The proofs of the following three assertions are easily derived from the previous observations, so we omit them.

Proposition 4.4. [85] Let V = (Vn)n∈N is a solid topological (resp. solid locally topological; closed solid locally topological) string in an arbitrary l-ideal F of a tvRs (E, C, t). Then there exists in (E, C, t) a string U = (Un)n∈N of the same type, such that U ∩ F = V . Proposition 4.5. [85] Let F be an arbitrary l-ideal in a tvRs (E, C, t) ¡ ¢ (j) T (j) and (V )j∈N = n∈N Vn j∈N a solid bornivorous σ -ultrabarrel in F in the induced topology. Then there exists a solid bornivorous σ -ultrabarrel (U (j)) = ¡ ¢ j∈N T (j) (j) (j) n∈N Un j∈N in (E, C, t) such that U ∩ F = V for each j ∈ N. Corollary. [85] The classes of ultra-(DF), countably quasiultrabarrelled, locally topological, ultra-b-barrelled and ultra-Db tvRs’s are stable with respect to arbitrary l-ideals.

3.5. 3SP IN THE CLASS OF LOCALLY CONVEX RIESZ SPACES

In this paragraph we shall study the three-space-problem for lcRs’s, as well as some additional properties concerning l-ideals and quotients of such spaces. When we speak about the short exact sequence of lcRs’s

j q (∗) 0 → (F,C ∩ F, t|F ) → (E, C, t) → (E/F, C/F, t/F ) → 0, we shall assume triple exactness, i.e.: 1◦ exactness in the vector sense, i.e. Im(j) = Ker(q); 2◦ topological exactness in the sense that j: F → E is a topological injection, and q: E → E/F is a topological homomorphism; 3◦ orders in F and E/F are canonical in connection with the order in E ; that means that we shall assume that F is an l-ideal of E such that in the quotient E/F the cone C/F = q(C) is proper and the order in F is given by the cone C ∩ F . The variety of properties in the category of locally convex Riesz spaces is enabled by the Dieudonn´etopology (see paragraph 3.2). As we have already ob- 0 served, each l-ideal F of the space (E, C, σs(E,E )) always carries the Dieudonn´e 0 0 topology, i.e. σs(E,E )|F = σs(F,F ), but the same is not always true for the quotient [78; Remark on p. 206]. As far as the three-space-problem is concerned, we have 3.5. 3SP in the class of locally convex Riesz spaces 93

Proposition 5.1. [88] If (∗) is a short exact sequence of lcRs’s and if t|F = 0 ◦ 0 σs(F,F ) and t/F = σs(E/F, F ), then t = σs(E,E ), i.e. the property of “having the Dieudonnétopology” is three-space stable. 0 Proof. First of all, observe that the topologies σs(E,E ) and t are comparable: 0 0 0 0 σs(E,E ) 6 t. Hence, σs(E,E )|F 6 t|F = σs(F,F ). As σs(F,F ) is the 0 0 coarsest locally solid topology which is finer than σ(F,F ), we have σs(F,F ) 6 0 0 σs(E,E )|F , too, and so σs(E,E )|F = t|F . 0 0 On the other hand, from σs(E,E ) 6 t it follows that σs(E,E )/F 6 t/F = ◦ ◦ 0 σs(E/F, F ). Since we always have σs(E/F, F ) 6 σs(E,E )/F [78], it follows that 0 0 σs(E,E )/F = t/F . According to Lemma II.4.1, we obtain that σs(E,E ) = t. For further investigations we shall need the following auxiliary assertion. Lemma 5.2. Let (F,C ∩ F, t|F ) be a closed l-ideal of an lcRs (E, C, t) and let (∗) be the corresponding short exact sequence. Then the mapping q lifts order- bounded subsets with closure, i.e. for each order-bounded subset A in E/F there exists an order-bounded subset B in E such that q(B) ⊃ A. Proof. Consider the dual spaces (E0,C0) and ((E/F )0, (C/F )0) = (F ◦,F ◦ ∩ 0 0 ◦ C ), the corresponding Dieudonnétopologies σs(E ,E) and σs(F , E/F ) and the transposed mapping qt: F ◦ → E0 . Let us prove that t ◦ ◦ 0 ◦ 0 0 0 q :(F ,F ∩ C , σs(F , E/F )) → (E ,C , σs(E ,E)) 0 ◦ ◦ is a topological injection, i.e. that σs(E ,E)|F = σs(F , E/F ). Since the Dieudonnétopology σs is the topology of uniform convergence on order-intervals, it will follow that q lifts order-bounded subsets with closure. First of all, as the q -image of each order-bounded subset of E is order-bounded 0 ◦ ◦ ◦ in E/F , we have σs(E ,E)|F 6 σs(F , E/F ). Conversely, σs(F , E/F ) is the coarsest locally solid topology which is finer than the weak topology σ(F ◦, E/F ). ◦ 0 ◦ 0 ◦ Also, σ(F , E/F ) = σ(E ,E)|F and σs(E ,E)|F is a locally solid topology on ◦ ◦ ◦ 0 ◦ F which is finer than σ(F , E/F ). Hence, σs(F , E/F ) 6 σs(E ,E)|F and the lemma is proved. Remark. The previous assertion can also be proved by the given method in some cases without the assumption of local-convexity of the spaces concerned, provided sufficiently rich dual spaces exist. A similar remark applies for some of the propositions that follow, too. We shall keep to the locally convex case. As we have seen, the property of “being order-quasibarrelled” is preserved when passing from an lcRs to its arbitrary quotient, but not always when passing to its closed l-ideal, which is analogous to the corresponding situation for the property of “being quasibarrelled” among locally convex spaces. When the three- space-problem is concerned, without additional assumptions it has the negative answer in the category of locally convex spaces (paragraph 2.4). However, we shall prove Theorem 5.3. [88] Let (∗) be a short exact sequence of lcRs’s, where F and E/F are order-quasibarrelled. Then E is an order-quasibarrelled lcRs, too. 94 III. Ordered topological vector spaces

Proof. Let T be a barrel in E which absorbs all order-bounded subsets. Then T ∩ F is a barrel in F which absorbs all order-bounded subsets, since each order- bounded subset in (F,C ∩ F ) is also an order-bounded set in (E,C). Therefore there exists a neighbourhood U of the origin in E such that T ⊃ T ∩F ⊃ (3U)∩F . Now, q(T ∩ U) is a barrel in E/F which absorbs all order-bounded subsets, according to the Lemma 5.2, and so, by the assumption, q(T ∩ U) is a neighbourhood of the origin in E/F . Further procedure is the same as in the proof of the Proposition II.4.3 — for the set V = U ∩ T ∩ U + F one can prove that it is a neighbourhood of the origin in E and that V ⊂ 2T , and so T is a neighbourhood of the origin in E , too. Order-countably quasibarrelled spaces were defined in paragraph 3.3. Du- ally speaking, an lcRs (E, C, t) is order-countably quasibarrelled if and only if each order-bornivorous barrel in it which is a countable intersection of closed t- neighbourhoods of the origin is a t-neighbourhood of the origin. We shall use the abbreviation COQ for such spaces [45]. With obvious changes one can prove Proposition 5.4. [88] Let (∗) be a short exact sequence of lcRs’s in which F and E/F are COQ. Then E is COQ; in other words, the property COQ is three-space stable. When order-(DF) lcRs’s (i.e. COQ Riesz spaces with a fundamental sequence of order-bounded subsets) are concerned, it was without detailed proof stated in [45] that the property is preserved when passing to an arbitrary quotient. A possible proof can be as follows: let (E, C, t) be an order-(DF) lcRs and F its closed l-ideal. 0 0 0 ◦ 0 ◦ ◦ Then (E ,C , σs(E ,E)) is a Fréchet lcRs, and so (F ,C ∩ F , σs(F , E/F )) is a 0 ◦ ◦ metrizable lcRs, since σs(E ,E)|F = σs(F , E/F ). It follows that the quotient (E/F, C/F ), as a Riesz space, possesses a fundamental sequence of order-bounded subsets. Since the property of “being COQ” is inherited by quotients, the space (E/F, C/F, t/F ) is order-(DF), too. When three-space stability of the mentioned property is concerned, again to the contrary of the non-ordered case, we have Theorem 5.5. [88] The property of “being order-(DF)” is three-space stable in the class of locally convex Riesz spaces. Proof. Let (∗) be a short exact sequence of lcRs’s in which F and E/F are order-(DF) spaces. From the previous proposition it follows that E is COQ. 0 0 ◦ ◦ Further, the spaces (F , σs(F ,F )) and (F , σs(F , E/F )) are Fréchet lcRs’s and 0 0 it should be mentioned that σs(F ,F ) = β(F ,F )(β —the strong topology). Also, β(F 0,F ) = β(E0,E)/F ◦ (Proposition 5.1). Now we have

0 0 ◦ 0 ◦ 0 0 σs(F ,F ) 6 σs(E ,E)/F 6 β(E ,E)/F = β(F ,F ) = σs(F ,F ), and so the following “dual sequence” is exact:

◦ 0 ◦ ◦ 0 0 0 0 0 0 0 0 → (F ,C ∩F , σs(F , E/F )) → (E ,C , σs(E ,E)) → (F ,C /F , σs(F ,F )) → 0. 3.5. 3SP in the class of locally convex Riesz spaces 95

0 0 0 According to [40; Theorem 6], (E ,C , σs(E ,E)) is a metrizable locally convex space, which means that (E, C, t) has a fundamental sequence of order-bounded subsets. In that way, it is proved that (E, C, t) is an order-(DF) space. The order-bound topology on a Riesz space (E,C) is the ﬁnest locally convex topology tbE for which every order-bounded subset in E is topologically bounded.

Proposition 5.6. [88] Let (∗) be a short exact sequence of lcRs’s in which t|F = tbF and t/F = tb(E/F ) . Then t = tbE , i.e. the property of “having the order-bound topology” is 3SP-stable.

Further, from t 6 tbE it follows that tb(E/F ) = t/F 6 tbE/F . On the other hand, tb(E/F ) is the finest locally convex topology for which all the order-bounded subsets are bounded, and so tbE/F 6 tb(E/F ) . Thus, tbE/F = tb(E/F ) = t/F . Using again Lemma II.4.1, we conclude that t = tbE . We have already seen that in some cases order structure provides more regular behaviour of “classical” topological vector properties. A reflexive, even semi- reflexive lcRs is always complete (see paragraph 3.2), so that in the class of lcRs’s there is no counterpart to K¯omura’s example of an incomplete Montel space. Properties of “being bornological” and “being quasibarrelled” are inherited by each l-ideal (see paragraph 3.3), which differs from the locally convex case without order. We shall use these facts to show that each quotient of a Montel (resp. Fréchet-Montel) lcRs is again of the same type. This means that in the class of lcRs’s the famous Köthe-Grothendieck Fréchet-Montel space whose quotient by some closed subspace is isomorphic to l1 does not exist. In other words, Pisier’s method cannot be used to construct counterexamples for non-three-space stability of certain properties (see also paragraph 3.3). Also, using the same facts we shall derive some conclusions in connection with distinguished and semi-reflexive lcRs’s. We have seen that semi-reflexivity in the class of locally convex spaces is inherited by arbitrary closed subspaces, but not by arbitrary quotients, even in the case of Fréchet spaces. However, we have

Proposition 5.7. [88] Let E be a FréchetlcRs and F its closed l-ideal. Then F and E/F are semi-reflexive if and only if E is semi-reflexive. Proof. Taking into account Proposition II.5.4, we only have to prove that semi-reflexivity of E implies semi-reflexivity of E/F . 0 0 From the semi-reflexivity of E it follows that Eτ = Eβ is a barrelled locally convex space, i.e. (E0,C0, β(E0,E)) = (E0,C0, τ(E0,E)) is a barrelled and also bornological lcRs (these two properties are in this case equivalent and equivalent to the property of being quasibarrelled [62], 29.4). Furthermore, F ◦ equipped 0 0 with the topology inherited from Eτ = Eβ is bornological (Theorem 3.2), and so 96 III. Ordered topological vector spaces quasibarrelled lcRs [137; p. 182]. Also,

◦ ◦ 0 ◦ 0 ◦ σ(F , E/F ) 6 σs(F , E/F ) = σs(E ,E)|F 6 τ(E ,E)|F = β(E0,E)|F ◦ 6 τ(F ◦, E/F ) 6 β(F ◦, E/F ), and so (F ◦,C0 ∩ F ◦, τ(F ◦, E/F )) is a bornological (quasibarrelled) lcRs, hence a barrelled lcRs (because the quotient E/F is Fréchet [62], 29.4). Therefore the space E/F is semi-reflexive. Proposition 5.8. [88] Each quotient E/F of a Montel (Fréchet-Montel)lcRs is a Montel (Fréchet-Montel)lcRs. Proof. Since E/F is (in both cases) a barrelled lcRs, to show that it is Montel ◦ 0 it is enough to prove that topologies β(F , E/F ) and (E/F )c (topology of uniform convergence on compact absolutely convex subsets of E/F ) are equal on F ◦ . Since 0 0 0 0 0 Ec = Eβ ,(E ,C , β(E ,E)) is a barrelled (and so quasibarrelled) lcRs. According to [137; p. 182], (F ◦,C0 ∩ F ◦, β(E0,E)|F ◦) is a quasibarrelled lcRs. Furthermore, 0 ◦ ◦ ◦ ◦ Ec|F 6 c(F , E/F ) 6 τ(F , E/F ) 6 β(F , E/F ) (Proposition I.6.12),

0 ◦ 0 ◦ 0 ◦ ◦ ∗ ◦ Ec|F = τ(E ,E)|F = β(E ,E)|F 6 τ(F , E/F ) 6 β (F , E/F ) and

(F ◦,C0∩F ◦, β(E0,E)|F ◦) = (F ◦,C0∩F ◦, τ(F ◦, E/F )) = (F ◦,C0∩F ◦, β∗(F ◦, E/F ))

0 ∗ ◦ ∗ and so (E/F )c = β (F , E/F ), where β is the topology of uniform convergence on strongly bounded subsets. Since weakly and strongly bounded subsets in E/F (which is barrelled) are the same, we have that β∗(F ◦, E/F ) = β(F ◦, E/F ) and the proposition is proved. When the property of distinguishedness is concerned, let us remark the following: In the class of Fréchet locally convex spaces (paragraph 2.5) there exists a short exact sequence of even reflexive spaces, with the mapping q which does not lift bounded sets. On the other hand, if q lifts bounded sets, then the property of “being distinguished” is three-space stable. However, in the class of Fréchet lcRs’s distinguishedness of the middle term in a short exact sequence (provided F and E/F are distinguished) is equivalent with the lifting of bounded sets of the mapping q . Moreover, if (E, C, t) is a distinguished Fréchet lattice, then for each closed l-ideal F , the respective quotient mapping q : E → E/F lifts bounded subsets. In contrast to the Fréchet locally convex spaces, in the class of Fréchet lcRs’s we can prove Proposition 5.9. [88] In the class of FréchetlcRs’s the property of “being distinguished” is inherited by every closed l-ideal and quotient. Proof. Let E in the sequence (∗) be a Fréchet lcRs. As in the proof of ◦ 0 ◦ 0 ◦ Proposition 5.7, (F ,C ∩ F ,Eβ|F ) is a quasibarrelled lcRs and also barrelled, 3.5. 3SP in the class of locally convex Riesz spaces 97 i.e. β(E0,E)|F ◦ = β(F ◦, E/F ). It is a consequence of the fact that in the dual space of an arbitrary barrelled space, a topology which is between the weak and the strong topology, is barrelled if (and only if) it is quasibarrelled, and in that case it is equal to the strong topology. Thus, E/F is a distinguished Fréchet lcRs. From the equality of topologies β(E0,E)|F ◦ and β(F ◦, E/F ) it follows that q lifts bounded sets (with closure). Furthermore, if the space E is distinguished, then, on the base of the Proposition II.5.9, the dual sequence

0 0 0 0 → (E/F )β → Eβ → Fβ → 0

0 is exact, and so Fβ is a barrelled space, i.e. F is distinguished. 0 0 0 Let now (E, C, t) be an lcRs and (E ,C , σs(E ,E)) the corresponding dual 0 lcRs equipped with the Dieudonn´etopology. The topology σs(E ,E) is in general not compatible with the duality hE,E0i. If E is not an l-ideal in E0b , then 0 0 0 0 (E ,C , σs(E ,E)) 6= E . Therefore we shall say that (E, C, t) is an order-semi- 00 0 0 0 0 reﬂexive lcRs if E|σ| = (E ,C , σs(E ,E)) = E . Similarly, we say that (E, C, t) 00 0 is order-distinguished if each σ(E|σ|,E )-bounded set is contained in the closure of some order-interval from E , i.e.

00 0 00 0 σ(E|σ|,E ) (∀A ∈ B(E|σ|,E ))(∃x ∈ C ∩ E) A ⊂ [−x, x] .

Let us state some properties of the introduced classes of spaces. 1◦ If E is an order-distinguished lcRs, then each bounded subset A of E is order-bounded. In fact,

σ(E00 ,E0) A = A ∩ E ⊂ [−x, x] |σ| ∩ E = [−x, x]E = [−x, x].

It follows that each order-distinguished lcRs is distinguished. 2◦ There exist distinguished spaces (even among Banach lattices) which are not order-distinguished. An example can be the space c0 with canonical order. Namely, it is known that order-intervals in c0 are compact disks, and so σ(l1, c0) 6 σs(l1, c0) 6 c(l1, c0) 6 τ(l1, c0). If c0 were order-distinguished, it would follow that σs(l1, c0) = τ(l1, c0) = β(l1, c0), i.e. in c0 bounded subsets would be order-bounded, and so each bounded set would be compact, and c0 would be finite-dimensional. ◦ 0 0 0 3 (E, C, t) is order-distinguished if and only if (E ,C , σs(E ,E)) is order- quasibarrelled, i.e. barrelled. 0 0 0 In fact, it follows from the definition that (E ,C , σs(E ,E)) is barrelled if and only if E is order-distinguished. But, since each order-interval [−x0, x0], x0 ∈ C0 ∩E0 in E0 is strongly bounded, (E0,C0, β(E0,E)) is an lcRs and order-bounded 0 0 0 subsets in it are topologically bounded. That is why (E ,C , σs(E ,E)) is order- quasibarrelled if and only if it is barrelled. Analogously, it is COQ if and only if it is countably barrelled. 4◦ The notions “order-distinguished” and “order-semi-reflexive” depend only on the dual pair hE,E0i, as well as the notions distinguished and semi-reflexive in the class of locally convex spaces. 98 III. Ordered topological vector spaces

Passing to the behaviour of the introduced properties in connection with the short exact sequence (∗), we shall prove Proposition 5.10. [88] 1◦ In the short exact sequence (∗) of FréchetlcRs’s let E be order-distinguished. Then E/F is also order-distinguished, while F need not have this property. 2◦ If the spaces F and E/F in the short exact sequence (∗) of Fréchet lcRs’s are order-distinguished, then E has the same property; hence, order- distinguishedness is a three-space stable property in the class of FréchetlcRs’s. Proof. 1◦ Firstly, observe that the space m is an order-distinguished Fréchet lcRs, because its bounded subsets are order-bounded, while its l-ideal c0 is not, as already mentioned. Let E be an order-distinguished Fréchet lcRs and F its closed l-ideal. As 0 0 0 0 stated before, β(E ,E) = σs(E ,E) and (E , σs(E ,E)) is a barrelled, which means 0 ◦ ◦ here also bornological, locally convex space. Since σs(E ,E)|F = σs(F , E/F ), ◦ 0 ◦ ◦ the space (F ,C ∩ F , σs(F , E/F )) is bornological, too, and so it is a quasibarrelled lcRs, and it also has to be barrelled. 2◦ Let now the spaces F and E/F in the short exact sequence (∗) of Fréchet lcRs’s be order-distinguished, which means also distinguished. Then in their duals the Dieudonnéand the strong topologies coincide. But then the mapping q lifts bounded sets because a subset of E/F is bounded if and only if it is order- bounded. Thus, E is a distinguished space (Corollary 2 of Proposition II.5.1) and the sequence 0 0 0 0 → (E/F )β → Eβ → Fβ → 0 is exact (Proposition II.5.9), i.e. the sequence

0 0 0 0 → (E/F )s → Eβ → Fs → 0

0 0 is exact. Taking into account Proposition 5.1, we obtain that Eβ = Es , and so the space E is order-distinguished.

Observe that the already mentioned example of spaces c0 and m shows that in the class of lcRs’s there is no matching result to the Proposition II.5.9:

j q 0 → c0 → m → m/c0 → 0 is a short exact sequence of Fr´echet lcRs’s (with norm-topologies and canonical orders), q lifts order-bounded subsets, but the dual sequence

0 0 0 0 0 0 0 → ((m/c0) , σs((m/c0) , m/c0)) → (m , σs(m , m)) → (c0, σs(c0, c0)) → 0 is not exact. Finally, for order-semi-reﬂexive spaces we have: Proposition 5.11. [88] If (E, C, t) is an order-semi-reﬂexive lcRs, then every closed l-ideal F in it and every quotient E/F are of the same kind. 3.5. 3SP in the class of locally convex Riesz spaces 99

Proof. The assertion about l-ideals follows from the fact that each order- interval in F is also an order-interval in E , which means weakly compact in E and also weakly compact in F (as a locally convex space). Since, by order-semi- 0 0 0 reﬂexivity of E , σ(E ,E) 6 σs(E ,E) 6 τ(E ,E), we obtain that

◦ ◦ 0 ◦ 0 ◦ ◦ σ(F , E/F ) 6 σs(F , E/F ) = σs(E ,E)|F 6 τ(E ,E)|F 6 τ(F , E/F ), i.e. E/F is an order-semi-reﬂexive lcRs. We do not know whether order-semi-reﬂexivity is a three-space stable property. IV. SUPPLEMENT: ENLARGEMENTS OF LOCALLY CONVEX TOPOLOGIES

We know that if an lcs (E, t) belongs to any of the four basic classes (barrelled, bornological, quasibarrelled, ultrabornologique), then it is endowed with the Mackey topology τ(E,E0), where the topological dual E0 is a vector subspace of the algebraic dual E∗ . If F is a dense hyperplane of E , we have investigated whether the space (F, τ(F,E0)) belongs to some of the basic classes. In a way a dual question can be the following: If the space (E0, τ(E0,F )) has a certain property P , does the space (E0, τ(E0,E)) have to possess the same property? Or, in the terms of the space E : If (E, τ(E,E0)) is a Mackey lcs with the property P , does the new Mackey topology τ(E,G), where G is a certain enlargement of the dual E0 , remain with the same property?

4.1. FINITE-DIMENSIONAL ENLARGEMENTS

Let E be a vector space over the ﬁeld K and let E∗ be its algebraic dual. Then a subspace F of E∗ forms a dual pair with E if and only if it is σ(E∗,E)- dense. Obviously then each subspace G of E∗ containing F also forms a dual pair with E . All such subspaces G have the form G = F +span A, A∩F = ∅ and they are called enlargements of F in E∗ . The dimension of the subspace span A is called the dimension of the enlargement. Some natural enlargements of the subspace F are the spaces F˜ (resp. F˙ ; F˜ ) of all linear forms which are bounded on σ(E,F )- bounded (resp. β(E,F )-bounded; σ(E,F )-compact) absolutely convex subsets; they are called the bounded (resp. strongly bounded; compact-bounded) closure of F . The space F is said to be boundedly (resp. strongly-boundedly; compact- boundedly) closed if F = F˜ (resp. F = F˙ ; F = F˜ ) is valid. For each subset F of E∗ and its natural enlargements we have

F ⊂ F˜ ⊂ F˙ ⊂ F.˜

The locally convex topologies σ(E,G), τ(E,G) and β(E,G), G = F +span A are called, respectively, enlargements of the weak σ(E,F ), Mackey τ(E,F ) and the strong topology β(E,F ). If the space E endowed with some of the topologies 4.1. Finite-dimensional enlargements 101

σ(E,F ), τ(E,F ) and β(E,F ) has a property P , it is natural to ask whether E with the enlarged topology σ(E,G), τ(E,G) or β(E,G) possesses the same property. In this chapter we give primarily results concerning the question of the inheritance of the barrelledness property from the space (E, τ(E,F )) to the space (E, τ(E,G)) which were obtained in the last twenty years. We start with natural extensions and the basic classes of spaces. If (E, t) is an arbitrary lcs with the topological dual E0 , then the natural extensions E˜0 (resp. E˙ 0 ; E˜0 ) are in the close relationship with the characterization of the basic classes of spaces. Namely, the following proposition can be proved routinely.

Proposition 1.1. An lcs (E, t) is bornological (resp. bornological-barrelled; ultrabornologique) if and only if t = τ(E,E0) and E0 = E˜0 (resp. E0 = E˙ 0 ; E0 = E˜0 ). We say that an lcs (E, t) is boundedly (resp. strongly-boundedly; compact- boundedly) closed if E0 = E˜0 (resp. E0 = E˙ 0 ; E0 = E˜0 ) holds. We shall state now a result obtained by Mackey which, together with the Dieudonn´e’sresult from [34] can serve as a motivation for considering enlargements of topologies.

Proposition 1.2. (G. W. Mackey [68]) The class of boundedly closed lcs’s is stable with respect to ﬁnite-dimensional enlargements. Proof. See [68] or [34], Theorem 1. Remark. If in the proof of the previous proposition one substitutes the set of all barrels of the space (E0, σ(E0,E)) with the set of all bornivorous barrels, then it is obtained that the class of strongly-boundedly closed lcs’s is stable with respect to ﬁnite-dimensional enlargements. When compact-boundedly closedness is concerned, we have

Proposition 1.3. The class of compact-boundedly closed lcs’s is not stable with respect to ﬁnite-dimensional enlargements. We prove ﬁrst of all a lemma.

Lemma 1.4. If (E, t) is a quasibarrelled lcs and G = E0+span{u}, u ∈ E∗\E0 is a one-dimensional enlargement, then τ(E,E0)|u−1(0) = τ(E,G)|u−1(0). Proof. By the Corollary of the Proposition II.1.2, the topology β(G, E) induces on E0 the topology β(E0,E). Furthermore, E0 is a sequentially closed hyperplane in the space (G, β(G, E)), since because of the quasibarrelledness of (E, t), the strong dual (E0, β(E0,E)) is sequentially complete. Thus, the canonical projection of (G, β(G, E)) to the space (E0, β(E0,E)) is sequentially continuous, and so bounded, which means that for each β(G, E)-bounded subset A of G there exist a β(E0,E)-bounded subset B of E0 and a bounded ﬁnite-dimensional subset C of span{u} such that A ⊂ B+C . Since the space (E, t) is quasibarrelled, furthermore 102 IV. Enlargements of locally convex topologies we have that each β(G, E)-bounded subset is relatively σ(G, E)-compact, i.e. the space (E, τ(E,G)) is quasibarrelled, too. Since we have

A◦ ∩ u−1(0) ⊃ (B + C)◦ ∩ u−1(0) = B◦ ∩ u−1(0) for each σ(G, E)-compact absolutely convex subset A and the respective σ(E0,E)- compact subset B , we conclude that τ(E,G)|u−1(0) > τ(E,E0)|u−1(0). The converse is obvious. Remark. The previous Lemma generalizes a result of Dieudonn´efrom [34] using the ideas of J. H. Webb from private communication. Proof of the Proposition 1.3. Let (E, t) be an arbitrary ultrabornologique space satisfying E0 6= E∗ . By the Theorem II.1.6, there exists a dense hyperplane F in E such that (F, t|F ) is not an ultrabornologique space. Let G = E0 +span{u}, where F = u−1(0), u ∈ E∗ \ E0 be a one-dimensional enlargement of the dual E0 . By the Proposition 1.1, the space (E, t) = (E, τ(E,E0)) is compact-boundedly closed. Let us show that the space (E, τ(E,G)) does not possess this property. Indeed, if G = G˜ holded, then by the Proposition 1.1 the space (E, τ(E,G)) would be ultrabornologique. Since by the previous Lemma τ(E,E0)|F = τ(E,G)|F , the space (F, t|F ) = (F, τ(E,E0)|F ) = (F, τ(E,G)|F ) would be ultrabornologique, because F is a τ(E,G)-closed hyperplane. Contradiction. Now we can formulate the following

Theorem 1.5. (a) The class of barrelled (resp. bornological; quasibarrelled) lcs’s is stable with respect to finite-dimensional enlargements. (b) The class of ultrabornologique lcs’s is not stable with respect to finite- dimensional enlargements. Proof. (a) From the proof of the Lemma 1.4 and using induction we conclude that the space (E, τ(E,G)) is quasibarrelled for each finite-dimensional enlargement G of the topological dual E0 , if the space (E, τ(E,E0)) is such. If in the same Lemma we substitute strong topologies by weak topologies σ(E0,E) and σ(G, E), we obtain a proof for barrelled spaces. The proof for bornological spaces follows directly applying Propositions 1.1 and 1.2. (b) Follows from Proposition 1.3. Using the mentioned Lemma and the connection between hyperplanes and linear forms we obtain Proposition 1.6. A basic class of lcs’s is stable with respect to dense hyperplanes if and only if it is stable with respect to one-dimensional enlargements. Proof. (E, τ(E,G)) = (F, τ(E,G)|F ) ⊕ (L, k k), where F = u−1(0), u ∈ E∗ \ E0 , G = E0 + span{u}, L is a one-dimensional normed subspace of E , L ∩ F = {0}. 4.2. Countable enlargements of barrelled topologies 103

4.2. COUNTABLE ENLARGEMENTS OF BARRELLED TOPOLOGIES

In this paragraph we shall suppose that (E, τ(E,E0)) is a barrelled lcs and that E0 6= E∗ ; further, M is a countable-dimensional subspace of E∗ such that M ∩ E0 = {0}. We investigate whether the space E with the “enlarged” Mackey topology τ(E,E0 + M) is again barrelled. If for some M the answer is positive, we say that the topology τ(E,E0 + M) is a barrelled countable enlargement (BCE) of the topology τ(E,E0). We shall show that for each barrelled lcs E with E0 6= E∗ there exists a countable enlargement which is not barrelled and we shall give several suﬃcient conditions for a space to have a BCE. The problem whether each barrelled space E has a BCE is still open. Most results of this and the following paragraph are due to W. J. Robertson, I. Tweddle, F. E. Yeomans and S. A. Saxon ([92], [111], [94], [100]). Obviously the subspace M can be represented as a union of an increasing ∗ sequence (Mn) of n-dimensional subspaces of E , while by the Theorem 1.5(a) it 0 follows that (τ(E,E + Mn))n∈N is an incresing sequence of barrelled topologies. We begin with some assertions which are of interest for obtaining properties of the enlarged topology τ(E,E0 + M). Proposition 2.1. Let (E, τ(E,E0)) be a barrelled lcs and let E0 + M be a countable-dimensional enlargement of the dual E0 . If A is a σ(E0 +M,E)-compact 0 0 absolutely convex subset of E + M , then A is contained in some E + Mn . 0 0 Proof. The sets An := A∩(E +Mn) are absolutely convex and σ(E +Mn,E)- 0 0 closed and bounded in E +Mn for each n ∈ N. Since the topology τ(E,E +Mn) 0 0 is barrelled, it follows that An is a σ(E + Mn,E)-compact subset of E + Mn ; 0 0 0 hence An is σ(E + M,E)-compact and so σ(E + M,E)-closed in E + M . Since the topology of the Banach space

[∞ [∞ [∞ 0 span A = nA = (A ∩ (E + Mn)) = nAn n=1 n=1 n=1

0 is finer than the weak topology σ(E +M,E)| span A, each An is a closed subspace of the Banach space span A. By the Baire’s category theorem, it follows that 0 A ⊂ nAn , i.e. A ⊂ E + Mn for some n ∈ N. This finishes the proof. Proposition 2.2. Let (E, τ(E,E0)) be a barrelled lcs. Then the space (E, τ(E,E0 + M)) is barrelled if and only if each σ(E0 + M,E)-bounded subset 0 0 of E + M is contained in some finite-dimensional enlargement E + Mn . Proof. Let (E, τ(E,E0 + M)) be a barrelled space. Then each σ(E0 +M,E)-bounded subset is τ(E,E0 +M)-equicontinuous and the result follows from the previous proposition. Conversely, suppose that each σ(E0 + M,E)-bounded subset A of E0 + M is 0 contained (and bounded) in σ(E + Mn,E) for some n ∈ N. Since the topology 104 IV. Enlargements of locally convex topologies

0 0 τ(E,E + Mn) is barrelled, it follows that A is also relatively σ(E + Mn,E)- compact and so relatively σ(E0 + M,E)-compact. This means that the topology τ(E,E0 + M) is barrelled. We are now in position to consider the completeness of the enlarged topology τ(E,E0 +M). We shall use A. Grothendieck’s completeness criterion (see [62], 21.9 or [103], IV.6): An lcs (E, t) is complete if and only if for each hyperplane H ⊂ E0 the fact that H ∩ A is σ(E0,E)-closed for every σ(E0,E)-closed absolutely convex t-equicontinuous subset A of E0 implies that H is σ(E0,E)-closed. Proposition 2.3. If (E, τ(E,E0)) is a barrelled lcs and G a finite or countable-dimensional enlargement of the dual E0 , then the space (E, τ(E,G)) is not complete. Proof. Each hyperplane H in G which contains E0 is obviously σ(G, E)-dense in G (since E0 is σ(E∗,E)-dense in E∗ ). If we show that H ∩ A is a σ(G, E)- closed subset in G for each τ(E,G)-equicontinuous absolutely convex σ(G, E)- closed subset A, this will prove that the space (E, τ(E,G)) is not complete. Let A be such a subset; then by the Proposition 2.2, A ⊂ G1 , where G1 is a finite- 0 dimensional enlargement of the dual E in G. Now if H ⊃ G1 , then A ∩ H = A 0 and the proof is complete. Otherwise, H∩G1 is a hyperplane in G1 . Since H ⊃ E , we have that (E, τ(E,H ∩ G1)) is a barrelled space by the Theorem 1.5(a). Like in the proof of Lemma 1.4, A ⊂ B + C , where B is a σ(H ∩ G1,E)-compact subset in H ∩ G1 and C is compact in the one-dimensional complement of H ∩ G1 in G1 . The weak closedness of the set A in B + C implies the weak closedness of the set A ∩ H ∩ G1 in (B + C) ∩ H ∩ G1 . In other words, A ∩ H is weakly closed in B , wherefrom it follows that A ∩ H is weakly closed in G. This proposition implies the following two elegant results. Corollary 1. If (E, τ(E,E0)) is a barrelled lcs (and E0 6= E∗ ), then the codimension of the dual E0 in E∗ is uncountable. Proof. If this is not the case, E∗ is a countable enlargement of the dual E0 and by the Proposition the space (E, τ(E,E∗)) is not complete. This is a contradiction since the finest locally convex topology τ(E,E∗) is always complete. Corollary 2. If (E, τ(E,E0)) is a complete barrelled lcs and H is a σ(E0,E)-dense subspace of E0 of finite or countable codimension, then the topology τ(E,H) is not barrelled. Proof. In the case that the topology τ(E,H) is barrelled, by the Proposition its enlarged Mackey toplogy τ(E,E0) is not complete. Contradiction. Now, using the previous Corollary, we can show that each barrelled topology τ(E,E0) has a non-barrelled countable enlargement. Theorem 2.4. If (E, τ(E,E0)) is a barrelled lcs with E0 6= E∗ , then there exists a countable-dimensional subspace M of E∗ such that E0 ∩ M = {0} and the topology τ(E,E0 + M) is not barrelled. 4.2. Countable enlargements of barrelled topologies 105

Proof. Let H be a hyperplane in E∗ containing E0 . By the Corollary 2 the topology τ(E,H) is not barrelled, which means that there exists a σ(H,E)- bounded subset B which is not τ(E,H)-equicontinuous. Also, the subset B is not τ(E,E0 + N)-equicontinuous for each ﬁnite-dimensional subspace N of E∗ , N ∩ E0 = {0}. Since the topology τ(E,E0 + N) is barrelled (because τ(E,E0) is such), by the Proposition 2.2, B is not contained in E0 + N . Further, this means that there is a linearly independent countable subset A of B such that E0 ∩ span A = {0}. Take M = span A. Subset A is then σ(E0 + M,E)-bounded, but it is not contained in any E0 + N , which by the Proposition 2.2 means that the topology τ(E,E0 + M) is not barrelled. The theorem is proved. In the sequel we consider conditions assuring that majority of barrelled spaces have BCE’s.

Proposition 2.5. Let (E, t) be an lcs in which there exists a t-bounded subset with the dimension of the linear span at least c. Then there exists a countable- dimensional subspace M of E∗ such that M ∩ E0 = {0} and each σ(E0 + M,E)- bounded subset B is contained in E0 + N , for some finite-dimensional subset N of M . Proof. See [111]. Remark. A great number of classes of lcs’s satisfy the conditions of the previous proposition. E.g. such classes are: – barrelled lcs’s with a fundamental sequence of bounded subsets and the dimension at least c; – barrelled metrizable lcs’s with the dimension greater than c; – infinite-dimensional Fréchet spaces; – barrelled spaces of the type C(X), where X is a Hausdorff completely regular topological space and C(X) is endowed with the topology of compact convergence; – ultrabornologique spaces with the topology distinct from the finest locally convex one. Since all these spaces are barrelled and satisfy the mentioned conditions (for details see [111]), by the Proposition 2.2, all of them have BCE’s. We shall contrast the previous proposition with an example of a barrelled space having a BCE, while all of its bounded subsets are finite-dimensional. We shall use the famous example of I. Amemiya and Y. K¯omura [4] of a dual pair hE,F i and a countable-dimensional subspace H of E∗ such that: 1◦ bounded subsets for the topologies σ(E,F ), σ(F,E) and σ(F + H,E) are all finite-dimensional; 2◦ F + H is σ(F + H,E)-separable. From 1◦ and 2◦ it follows that the spaces (E, σ(E,F )) = (E, τ(E,F )) and (E, σ(E,E + H)) = (E, τ(E,E + H)) are barrelled and the codimension of the subspace F in F +H is countable. The last assertion follows from the fact that F is 106 IV. Enlargements of locally convex topologies not σ(F,E)-separable and the statement [94] that a finite-codimensional subspace of a separable space is separable. Remark. The only class of infinite-dimensional metrizable barrelled spaces which is not covered by the Proposition 2.5 is the one whose spaces are of dimension c and the bounded subsets are of at most countable dimension. However, in [108] an example is constructed of a dense barrelled subspace ψ of ω (= RN or CN ) whose bounded subsets have at most countable dimension and both the dimension and codimension of ψ in ω are equal to c. It is proved that each c-dimensional subspace of ψ is barrelled in the induced topology, that its closure is of finite codimension and that the topology σ(ψ, ϕ) has a BCE. Using the properties of the space ψ , the following result can be proved [108]: Proposition 2.6. Let (E, τ(E,E0)) be a barrelled lcs of the dimension c and suppose that it has a c-dimensional bounded subset. Then there exists a countable- dimensional subspace M of E∗ such that E0∩M = {0}, τ(E,E0+M) is a barrelled topology and σ(E,E0 + M)-bounded subsets have at most countable dimension. Remark. The first part is covered by the Proposition 2.5. The second part shows that there exists a BCE in which bounded subsets are of “less” dimension. The following proposition is of interest. Proposition 2.7. If an lcs (E, τ(E,E0)) has an infinite-dimensional bounded subset, then each finite or countable-dimensional enlargement τ(E,G) of the topology τ(E,E0) has an infinite-dimensional bounded subset. Proof. If each τ(E,G)-bounded subset is finite-dimensional, then the topology σ(G, E) is barrelled. But then by the Theorems II.1.1 and II.1.9 it follows that the topology σ(E0,E) is barrelled, which is equivalent with the fact that all τ(E,E0)- bounded subsets are finite-dimensional. Contradiction.

4.3. SATISFACTORY SUBSPACES OF BARRELLED SPACES

Let again (E, τ(E,E0)) be a barrelled lcs with E0 6= E∗ . We shall consider now some properties of the enlarged topology τ(E,E0 + M) in terms of some subspaces of E . If M is a countable-dimensional subspace of E∗ , then its polar M ◦ in E is of countable or uncountable codimension. Indeed, M = span{f1, f2,... } for ◦ some linearly independent sequence (fn). Then M = { x ∈ E | f(x) = 0 for all T∞ −1 ◦ 0 x ∈ M } = n=1 fn (0), i.e. M is an intersection of τ(E,E )-dense hyperplanes. Tk −1 Taking Fk = n=1 fn (0), we have

◦ E ⊃ F1 ⊃ F2 ⊃ · · · ⊃ Fk ⊃ Fk+1 ⊃ · · · ⊃ M .

Moreover, it is Fk 6= Fk+1 for each k since the vectors f1 , ... , fk+1 are linearly independent. Recall that an lcs (E, t) is quasi-complete if each closed absolutely 4.3. Satisfactory subspaces of barrelled spaces 107 convex t-bounded subset in it is complete. Besides, a Mackey space (E, τ(E,E0)) is barrelled if and only if the space (E0, σ(E0,E)) is quasi-complete, i.e. if and only if E0 is σ(E∗,E)-quasi-complete. If this is not the case, we shall denote by E0∼ the smallest σ(E∗,E)-quasi-complete subspace containing E0 . Then the topology τ(E,E0 + M) is barrelled if and only if E0 + M = (E0 + M)∼ . The inclusions E0 + M ⊂ E0 + M ∼ ⊂ (E0 + M)∼ always hold. If a locally convex topology t on E is not barrelled, then there exists the coarsest barrelled topology on E ﬁner than t (denoted as t∼ ) which is called the associated barrelled topology for t. Remark that the dual of the associated barrelled space (E, t∼) is exactly E0∼ (i.e. σ(E∗,E)-quasi-completion of the dual E0 ). For details see [61], [104]. Since we have seen that M ◦ is of countable or uncountable codimension in E , we start with the following

Proposition 3.1. If M ◦ has a countable codimension in E , then the topology τ(E,E0 + M) is not barrelled. Proof. Since M ◦ is a σ(E,E∗)-closed subspace of E and dim E/M ◦ = codim M ◦ , we have σ(E∗,E)|M ◦◦ = σ(M ◦◦, E/M ◦). Further, M ◦◦ , and so also M is a metrizable space in the weak topology, and hence M ∼ = M ◦◦ = completion of M , but M ∼ 6= M (there does not exist a complete metrizable space of countable dimension). If we suppose that the topology τ(E,E0 + M) is barrelled, we have that E0 + M ⊂ E0 + M ∼ ⊂ (E0 + M)∼ = E0 + M, wherefrom it follows that M ∼ ⊂ E0 +M . Thus, because of M ⊂ M ∼ , the following equality holds: M ∼ = (M ∼ ∩ E0) + M. As far as M ∼ is a Fr´echet space and E0 is σ(E∗,E)-quasicomplete (the barrelledness of the Mackey topology τ(E,E0) is supposed), we have that M ∼ ∩ E0 is σ(E∗,E)-quasicomplete, metrizable, which means complete, i.e. closed. We have obtained that a Fr´echet space M ∼ has a closed vector subspace M ∼ ∩ E0 of countable codimension. Contradiction.

Proposition 3.2. If M ◦ is dense and barrelled in the lcs (E, τ(E,E0)), then τ(E,E0 + M ∼) is the barrelled topology associated to τ(E,E0 + M). Proof. See [92], p. 389.

Proposition 3.3. If M ◦ is dense and of countable codimension in the lcs (E, τ(E,E0)), then τ(E,E0 + M) is not barrelled and its associated barrelled topology is τ(E,E0 + M ◦◦). Proof. That the topology τ(E,E0 +M) is not barrelled follows from the Propo- sition 3.1. The corresponding proof also shows that under the given assumptions M ∼ = M ◦◦ , which, together with the previous proposition implies the second statement. 108 IV. Enlargements of locally convex topologies

The following proposition gives a condition, suﬃcient for the assumptions of the previous one.

Proposition 3.4. If in a barrelled space (E, τ(E,E0)) there exists a bounded subset of inﬁnite dimension, then there exists a subspace M of E∗ , M ∩ E0 = {0} such that M ◦ is dense and of countable codimension in E . Proof. See [92], p. 390. Remark. The Proposition 3.1 implies that if a barrelled space (E, τ(E,E0)) has a BCE τ(E,E0 + M), then the codimension of the subspace M ◦ in E is uncountable. By an example it was shown in [92], pp. 392–393 that the converse is not true. Namely, in that example M ◦ is dense, barrelled and of uncountable codimension, and still τ(E,E0 + M) is not a barrelled topology. Before stating one of the important results, we give a technical lemma.

Lemma 3.5. Let (E, t) and (G, p) be lcs’s, (F, t|F ) a dense barrelled subspace of (E, t) and A a set of t-p continuous linear mappings from E to G. If A is pointwise bounded on (F, t|F ), then A is equicontinuous on (E, t). Proof. The barrelledness of the subspace (F, t|F ) and the pointwise boundness of the set A imply equicontinuity of A on (F, t|F ), i.e. for each p-closed absolutely convex p-neighbourhood of the origin V in G there exists a t-open neighbourhood of the origin U in E such that f(U ∩ F ) ⊂ V for each f ∈ A. Then we have t U ⊂ U ∩ F because (F, t|F ) is a dense subspace of (F, t). It follows that f(U) ⊂ p f(U ∩ F ) ⊂ V = V for each f ∈ A. The proof is complete.

Theorem 3.6. Let a barrelled lcs (E, τ(E,E0)) has a dense barrelled subspace of codimension at least c. Then it has a BCE. Proof. Let F be a vector subspace of E containing the given subspace and whose codimension is c. Such a subspace obviously exists and it is barrelled and dense in (E, τ(E,E0)). Let, further, G be the algebraic complement of F in E which is obviously algebraically isomorphic with KN ; let H be the topological dual of G with respect to the product topology and M the set of all extensions to E of elements from H , deﬁned to be equal to 0 on F . Then M ∩ E0 = {0} since F is dense in E , M ◦ = F and M is of countable dimension. Let now B be a σ(E0 + M,E)-bounded subset. Then by the Lemma 3.5, the projection X of B to E0 is σ(E0,M ◦)-bounded, i.e. τ(E,E0)-equicontinuous, since M ◦ is dense and barrelled in E . The projection Y of the set B to M is bounded and of ﬁnite dimension, since σ(H,G)-bounded subsets are such. Thus, B ⊂ X + Y is τ(E,E0 +M)-equicontinuous, i.e. the Mackey topology τ(E,E0 +M) is barrelled. In the sequel dense and barrelled subspaces with the codimension at least c will be called satisfactory.

Proposition 3.7. Let (E, τ(E,E0)) be a barrelled lcs and (F, τ(E,E0)|F ) its arbitrary barrelled subspace. Then, if F has a satisfactory subspace, so has E . 4.3. Satisfactory subspaces of barrelled spaces 109

Proof. Let G be the algebraic complement of the subspace F and let F1 be a satisfactory subspace of F . Put E1 = F1 + G. The subspace E1 is obviously 0 0 τ(E,E )-dense and of dimension at least c. Let us prove that (E1, τ(E,E )|E1) is barrelled. So, let T be a barrel in E1 . Then T ∩ F1 is a barrel in F1 and so a neighbourhood of the origin. Since F1 is dense in F , we have

τ(E,E0) E1 ⊂ span T ∩ F1 ⊂ span T .

τ(E,E0) Thus, T is a barrel in the space (E, τ(E,E0)), i.e. a τ(E,E0)-neighbourhood τ(E,E0) of the origin, wherefrom it follows that T = T ∩ E1 is a neighbourhood of the origin in E1 . Corollary. If (E, t) is a strict inductive limit of the sequence of barrelled 0 spaces (En, τ(En,En)) and if a term of the sequence has a satisfactory subspace, then so has the space (E, t). The following example shows a way of “making” of satisfactory subspaces. Example. Let (E, τ(E,E0)) be a barrelled lcs with E0 6= E∗ ,(F, τ(E,E0)|F ) a dense hyperplane and Λ an index set of cardinality at least c. Putting Eλ = E , L 0 Fλ = F for each λ ∈ Λ, then λ∈Λ(Fλ, τ(Fλ,Fλ)) is a satisfactory subspace of L 0 λ∈Λ(Eλ, τ(Eλ,Eλ)). A lemma of algebraic character follows.

Lemma 3.8. Let E be a vector space of dimension at least c. Then it can S∞ be represented as E = n=1 En , where (En) is an increasing sequence of vector subspaces of E , each having the codimension at least c. S Proof. If { xλ | λ ∈ Λ } is a base in E , then Λ = n>1 Λn , where (Λn) is an increasing sequence and |Λn| = |Λ \ Λn| = |Λ| for each n ∈ N. Let, furthermore, En = span{xλ | λ ∈ Λn }. Each En is obviously of codimension |Λ| > c. Corollary 1. If (E, τ(E,E0)) is a barrelled lcs with dim E > c and its completion (E,ˆ τˆ(E,Eˆ 0)) is a Baire space, then E has a dense subspace of codimension at least c. S Proof. Take E = n En as in the Lemma, and some En will be dense by [113], Theorem 4.

Corollary 2. Each Baire space (E, t) with dim E > c has a satisfactory subspace. S Proof. By the Lemma and the previous Corollary, E = n>m En , where Em and each En , n > m are dense in E . Then for some n > m, En is of second category and so barrelled.

Corollary 3. The strict inductive limit of a sequence of Fr´echetspaces has a satisfactory subspace. 110 IV. Enlargements of locally convex topologies

Proof follows by the previous and the Corollary of Proposition 3.7. Remark. If (E, t) is an lcs with dim E > c, then it has a satisfactory subspace if it obeys the property: S∞ (db) If (E, t) = n=1 En ,(En) is an increasing sequence of subspaces, then (Em, t|Em) is barrelled and dense in E for some m ∈ N. Since it is not known whether each lcs (E, t), E0 6= E∗ has a satisfactory subspace, hence a BCE, it is natural to consider the class of spaces with this property. So, it is said that an lcs (E, t) is fit if it contains a dense subspace (F, t|F ) with the codimension as large as possible (codimE F = dim E ). (E, t) is said to be barrelledly fit if it contains a dense barrelled subspace (F, t|F ) with codimE F = dim E . Trivially, an lcs (E, t) is barrelledly fit if and only if it contains a dense barrelled subspace (F, t|F ) with dim F = codimE F . It follows:

Proposition 3.9. If (E, t) is a barrelledly fit lcs with dim E > c, then it contains a satisfactory subspace. The following example shows that the converse is not true. Example. Let (E, t) be a space of dimension c with a satisfactory subspace (F, t|F ) and let (G, τ(G, G∗)) be a space of dimension d > c with the finest locally convex topology. Then (F, t|F ) × (G, τ(G, G∗)) is a satisfactory subspace of the space (E, t) × (G, τ(G, G∗)). The product space so obtained is not barrelledly fit. Indeed, if L is a dense subspace of E×G, then its projection is dense in G, i.e. equal to G (because G has the finest locally convex topology). Hence (E × {0}) + L = E × G, and so codim L 6 dim E = c < d = dim(E × G), i.e. (E, t) × (G, τ(G, G∗)) is not fit.

Proposition 3.10. The class of barrelledly fit lcs’s is stable with respect to arbitrary products and direct sums. Proof is straightforward. Example. The class of barrelledly fit lcs’s is not stable with respect to quotients. Indeed, let E 6= {0} is barrelledly fit in some locally convex topology t. Then (E, t) × ϕ is barrelledly fit, but it has ϕ as a quotient which is not fit (ϕ is the countable-dimensional vector space with the finest locally convex topology). The product {0} × ϕ further shows that the class of barrelledly fit spaces is not stable with respect to closed subspaces. Also, the previous example shows that a space may have a barrelledly fit quotient and a nontrivial barrelledly fit closed subspace without being barrelledly fit. As a contrast, we have:

Proposition 3.11. The class of barrelledly ﬁt lcs’s is three-space-stable.

Proposition 3.12. The class of barrelledly ﬁt lcs’s is stable with respect to countable-codimensional subspaces. Proof. See [100]. 4.3. Satisfactory subspaces of barrelled spaces 111

It is clear immediately that an lcs (E, t) is fit if the space (E, σ(E,E0)) is such. In the paper “The codensity character of topological vector spaces” (announced by S. A. Saxon in [100]) it is proved that a space is fit if it has a base of neighbourhoods of the origin whose cardinality is not greater than the dimension of the space. It follows that each infinite-dimensional metrizable space is fit. We give the following fitness criterion. Proposition 3.13. If an lcs (E, t) is of infinite dimension and dim E > dim E0 , then it is fit. Proof. Let d be the dimension of the space E and let G be a collection of d subspaces of dimension d whose algebraic direct sum is E (see [93], Example (d)). By the Hahn-Banach theorem, for each G ∈ G

dim G0 6 dim E0 6 d < 2d = dim G∗, wherefrom it follows that G0 6= G∗ . Hence, for each G ∈ G there exists a one- dimensional subspace G0 ⊂ G, such that µ [ ¶ E0 := span G0 G∈G is a dense d-dimensional subspace of (E, t), and so the space (E, t) is fit. Theorem 3.14. Assume the continuum hypothesis. Let, further, (E, τ(E,E0)) be a barrelled lcs containing a subspace M with dim M > c > dim M 0 . Then E contains a dense barrelled subspace whose codimension is equal to dim M . In particular, E contains a satisfactory subspace. Proof. See [100]. Corollary 1. Assuming the continuum hypothesis, if (E, τ(E,E0)) 6= ϕ is an infinite-dimensional barrelled space with dim E0 6 c, then it is barrelledly fit. Proof. Up to an isomorphism, ϕ is the only countable-dimensional barrelled space. Taking M = E in the Theorem, the assertion follows. Corollary 2. Assuming the continuum hypothesis, each barrelled space containing a separable subspace of uncountable dimension has a satisfactory subspace. Particularly, each infinite-dimensional separable barrelled space with E0 6= E∗ has a satisfactory subspace; moreover, it is barrelledly fit. Proof. The dual of a separable lcs has dimension not greater than c. Corollary 3. Assuming the continuum hypothesis, every separable infinite- dimensional barrelled space except ϕ has a BCE. Proof. By Corollary 2 and Theorem 3.6. Remark finally that the class of barrelled spaces with a BCE is stable with respect to countable-codimensional subspaces [21]. Also, a barrelled space has a 112 IV. Enlargements of locally convex topologies

BCE if some of its quotients is such [21]. The class of spaces having satisfactory subspaces is stable with respect to countable-codimensional subspaces, too, [100], Note on p. 1025. We ﬁnish remarking that when open problems 8, 9 and 10 are concerned (see the list in the next paragraph), a positive answer to the problem 10 would imply a positive answer to problem 9, and the latter would imply a positive answer to the basic problem number 8. However some recent results ([138], [139]) suggest that the answer to that question is probably negative.

4.4. OPEN PROBLEMS

We shall list in this paragraph some open problems mentioned in this book, or related to the concepts treated in it. 1. Is every infinite-dimensional Fréchet space regenerative? [98] 2. Does every infinite-dimensional Banach space possess a dense non-barrelled subspace? Equivalently: Does every infinite-dimensional Banach space have a separable quotient? [102] 3. Is the class of locally topological tvs’s stable with respect to finite-dimensional subspaces? [2] 4. Is the class of σ -barrelled lcs’s three-space-stable? [51]

5. Are the classes of B - and Br -complete spaces of Pták[63] three-space-stable? 6. Given a short exact sequence 0 → F → E → G → 0 of Banach spaces such that F and G are dual Banach spaces, is it then true that E is a dual Banach space and that the sequence is a dual sequence? [31] 7. Is the class of order-semi-reflexive lcRs’s three-space-stable? [88] 8. Does every barrelled space E with the topology distinct from the finest locally convex one possess a barrelled countable enlargement? [92] 9. Does every E as above have a satisfactory subspace? [94] 10. If an lcs (E, τ(E,E0)) is barrelled and fit, is it barrelledly fit? [100] REFERENCES

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absolute value 73 Fréchet 80 Banach disc 42 topological vector 79 band 76 vector 73 barrel 23 lcs 12 d- 47 lcRs 79 solid 84 l-homomorphism 76 σ- 47 l-ideal 76 canonical bilinear form 29 σ-normal 85 codimension 6 lifting of bounded sets 20, 63 cone 72 limit generating 72 (∗-) inductive 19 proper 72 projective 16 cover of a set strict (∗-) inductive 19 absolutely convex 3 local Cauchy sequence 57 circled 4 local zero sequence 57 closed (absolutely) convex 10 locally convergent sequence 57 convex 3 mapping order-convex 73 closed 28 solid 74 locally bounded 57 dimension 6 (∗-) direct sum 18 locally continuous 57 dual sequence 66 Minkowski functional 13 dual space negative part 73 algebraic 6 order-bounded form 77 order- 77 order-interval 73 order-bounded 77 Pisier method 68 topological 10 polar 30 dual system (duality) 29 positive form 77 enlargement 100 positive part 73 barrelled countable (BCE) 104 property F-norm 8 (DC) 66 hyperplane 6 extension 62 almost closed 50 lifting 62 ultradense 50 (S) 53 lattice Riesz decomposition 73 Banach 80 quotient 6 Subject index 121 saturated family 22 locally bounded 11 seminorm 12 locally convex 12 F- 8 locally convex Riesz 79 Riesz 79 locally order-complete 80 set locally topological 64 absolutely convex 3 Mackey 33 absorbing 4 Montel 54 bornivorous 26 of minimal type 8 bounded 11 order-bornological 85 circled 3 order-complete 75 convex 3 order-countably quasibarrelled 86 equicontinuous 23 order-(DF) 94 equivorous 55 order-distinguished 97 locally dense 57 order-semi-reflexive 97 order-bounded 73 order-sequentially quasibarrelled 86 order-convex 73 order-quasibarrelled 85 precompact 11 order-quasiultrabarrelled 90 solid 73 order-ultrabornological 90 strongly bounded 33 order-σ-quasibarrelled 86 solid kernel 74 ordered vector 72 space p- 48 associated bornological 41 pr- 48 b- 48 p-barrelled 48 b0- 48 quasibarrelled 26 barrelled 23 quasicomplete 34 barrelledly fit 110 quasiultrabarrelled 26 b-barrelled 48 regenerative 46 bornological 26 reflexive 35 boundedly closed 83 Riesz 73 boundedly order-complete 80 semi-reflexive 34 b-reflexive 55 sequentially barrelled 48 countably barrelled 47 sequentially quasibarrelled 48 countably quasibarrelled 48 topological vector 6 countably quasiultrabarrelled 58 topological vector Riesz 79 countably ultrabarrelled 58 totally reflexive 55 Db- 48 ultrabarrelled 23 (DF)- 48 ultra-b-barrelled 64 distinguished 66 ultrabornological 26 dual 66 ultrabornologiques 39 fit 110 ultra-(DF) 58 Fréchet (F-) 9 uniform 12 k- 48 weak Riesz 73 kr- 48 σ-locally topological 64 k-barrelled 48 σ-order-complete 76 linear topological 6 ω-barrelled 47 122 Subject index

ω-quasibarrelled 48 SOME NOTATIONS string 4 kernel of a 5 knot of a 5 co A,Γ A 3 topological 6 co A ,Γ A 3 strong bidual 35 i i i i E∗ 6 sublattice 76 t , t , t 7 subspace 6 F b(E) m(E) E0 10 normal 76 B(E), P(E) 11 satisfactory 108 p 12 Riesz 76 M t|F 15 theorem lim g (E ) 16 Alaoglu-Bourbaki 32 ←− ij j t/F 18 Banach-Steinhaus 24, 25, 27 L L E , ∗- E 18 bipolar 30 i∈I i i∈I i lim(E , t ), ∗-lim(E , t ) 19 closed graph 28 −→ i i −→ i i L (E,F ) 22 Hahn-Banach 14 M L (E,F ), L (E,F ) 22 Kolmogoroff 14 s c L (E,F ), L (E,F ) 22 Mackey-Arens 32 p b (E , p ) 28 open mapping 28 B B hx, yi, hF,Gi 29 three-space-problem 39 σ(F,G), σ(E,E0) 29 topology A◦ 30 associated barrelled 107 τ(F,G), τ(E,E0) 33 compatible 31 β∗(E,E0), β(E,E0) 34 Dieudonné81 E0 34 finest linear solid 90 β E00 35 (∗-) inductive 17 τ(E, E˜0) 41 linear 6 τ(E,E∗) 42 M- 21 ω 45 Mackey 33 ˜0 of compact convergence 22 τ(E, E ) 52 TE0 55 of precompact convergence 22 b b∗ β of simple convergence 22 t , t , t 56 of uniform convergence 21 [x, y], [A] 73 order-bound 95 sup(x, y), inf(x, y) 73 x+, x−, |x| 73 projective 15 d strong 22, 33 x ⊥ y, A 73 vector 6 SA, sk(A) 74 [C]F 76 weak 29 b ∗ tvs 6 E , C 76 C0 81 tvRs 79 0 0 ultrabarrel 23 σs(E,E ), σs(E ,E) 81 tbE 95 t∼ 110