The Topological and Bornological Approaches in Locally Convex Spaces

Home , Bornological space, Local boundedness, Polar topology, Vector bornology

The topological and bornological approaches in locally convex spaces

Ngai-Ching Wong Department of Applied Mathematics National Sun Yat-sen University Kao-hsiung, 80424, Taiwan email: [email protected]

Abstract This paper is devoted to binding two common techniques in functional analysis: the topological method and the bornological method. The usual approaches in the theory of locally convex spaces due to Grothendieck (via Banach space operators), Randtke (via continuous seminorms) and Hogbe-Nlend (via convex bounded sets) can thus be compared. Key words: operator ideals, locally convex spaces, topologies, bornologies, nuclear spaces, Schwartz spaces.

1 Introduction

The author learned from his late teacher Yau-Chuen Wong (1935.10.2–1994.11.7) the so-called working on two-legs’ philosophy. Quoted from [25], his “two legs” are topologies and bornologies. It is well-known that the continuity and (local) boundedness of a linear operator between normed spaces are equivalent. This is because the unit ball of a normed space simultaneously serves as a neighborhood of zero and a bounded set. It is, however, no longer true in the context of locally convex spaces. The Mackey-Arens Theorem indicates that topologies (families of neighborhoods) and bornologies (families of bounded sets) are in dual pair ... Following Grothendieck and Pietsch (cf. [13]), Yau-Chuen associated to each type of topologies or bornologies a special class of operators, that is, operator ideals in modern terminology [14]. By analyzing operator ideals, in particular their duality, Yau-Chuen discovered useful information of the topological and bornological structure of corresponding spaces (see e.g. [30, 27, 24]). Following Yau-Chuen’s ideal, we explore the equivalence among the topological and the bornological methods in studying locally convex spaces. This paper is a short version of [26], from which readers can ﬁnd detailed proofs of all results presented here. The following commutative diamond-like diagram summaries our study of locally convex spaces.

1 Operators A Ó << Ó ÓA ^<< < ÓÓ Ó < << Ó ÓÓ << < projective ÓÓ Ó < << inductive Ó ÓÓ << < topologies T ÓÓ Ó < << bornologies B Ó ÓÓ << < ÓÓ Ó < << Ó ÓÓ b b << < ÓÓ Ó O,O O,O < << Ó ÓÓ << < ÓÓ Ó < << Ó ÓÓ << < ÓÓ Ó ◦ < << ÑÓ ÓÓ polar P << TopologiesÓ P / Bornologies M o 44 polar M◦ 44 4 4 44 44 4 4 44 44 Randtke 4 Grothendieck Hogbe-Nlend 4 44 4 4 44 44 44 44 ÐØ Spaces The theory of operator ideal is founded by Pietsch [14], which is originated from the works of Grothendieck [4] and Schatten [17]. See also [12, 8, 10, 2] for more information. The idea of generating topologies and generating bornologies are due to Stephani [18, 19, 20, 21, 22] and Franco and Piñeiro[3] in the context of Banach spaces. The explicit construction (with all arrows shown above) of the triangle is given in [30], in which several applications to Banach space theory are demonstrated. When the underlying space is a fixed complex Hilbert space, West implements the triangle in the context of operator algebras [23] and provides several applications with Conradie [1]. In [27], we overcome the major difficulty in binding the bornological part to other parts of the triangle in the context of locally convex spaces. Namely, we establish a concrete construction of the bornologically surjective hulls of operator ideals on locally convex spaces. The concrete construction of injective hull of an operator ideal on locally convex spaces is given earlier in [3]. In [24], we show that in the study of locally convex spaces, the power of either the topological machinery of Randtke (via continuous seminorms) [15] or the bornological machinery of Hogbe-Nlend (via convex bounded subsets) [5, 6] is as strong as that of the operator theoretical machinery of Grothendieck (via Banach space operators) (see [28, 29] or [10]) (if one follows the implication arrows of the above diagram without going through any polar path). As a conclusion of above works, the paper [26] establishes the commutativity of the triangle (with polar paths) in the context of locally convex spaces and the equivalence among all three approaches of Grothendieck, Randtke and Hogbe-Nlend (and thus one can follow almost freely an arbitrary path of implication arrows with or without going through any of the polar paths).

2 Preliminaries

Throughout this paper, all vector spaces have the same underlying scalar field K, which is either the field R of real numbers or the field C of complex numbers. Let C be either the class LCS of locally convex (Hausdorff) spaces or the class of Banach spaces over the scalar field K. The following terminologies can be found in, for example, [30]. In particular, for every X and Y in C, we denote by Lb(X,Y ), L(X,Y ) and L×(X,Y ) the collection of all operators from X into Y which are bounded (i.e. sending a 0-neighborhood to a bounded set), continuous and locally bounded (i.e. sending bounded sets to bounded sets), respectively. Let X0 be the (topological) dual space of a LCS X. Denote by σ(X,X0) the weak topology of X, and

2 by Pori(X) the original topology of X. We employ the notion Mfin(Y ) for the finite dimensional bornology of Y which has a basis consisting of all convex hulls conv{y1, . . . , yn} of finite subsets {y1, . . . , yn} of Y . On the other hand, Mvon(Y ) is used for the von Neumann bornology of Y which consists of all topologically bounded subsets of Y . Ordering of topologies and bornologies are induced by set-theoretical inclusion, as usual. Moreover, we write briefly XP for a vector space X equipped with a locally convex topology P and Y M for a vector space Y equipped with a convex vector bornology M. The (absolute) polar B◦ of a subset B in X0 is defined by

B◦ = {x ∈ X0 : |hx, bi| ≤ 1, ∀b ∈ B}.

0 • 00 Whenever A ⊂ X , denote by A the polar of A taken in Xββ, namely,

• 00 A = {x ∈ Xββ : |ha, xi| ≤ 1, ∀a ∈ A},

00 ◦ where Xββ is the strong bidual of X, while A denotes the polar of A taken in X. 0 0 0 Let P(Xβ) be a locally convex topology and M(Xβ) a convex vector bornology of the strong dual Xβ ◦ 0 of X. The polar topology M (X) of M(Xβ) is determined by those open subsets V of X whose polar ◦ 0 ◦ 0 V is M(Xβ)-bounded. Similarly, The polar bornology P (X) of P(Xβ) is determined by those bounded ◦ 0 subsets B of X whose polar B is a 0-neighborhood in the P(Xβ) topology. A subset B of a LCS X is said to be a disk if B is absolutely convex, i.e., λB + βB ⊂ B whenever P∞ |λ| + |β| ≤ 1. B is said to be a σ–disk, or absolutely σ–convex if n=1 λnbn converges in B whenever P∞ the scalar sum n=1 |λn| ≤ 1 and bn ∈ B, n = 1, 2,... . A bounded disk B is said to be infracomplete S∞ if the normed space X(B) = n=1 nB equipped with the gauge γB of B as its norm is complete, where γB(x) = inf{λ > 0 : x ∈ λB} for each x in X(B). Bounded σ-disks are infracomplete, and continuous images of σ-disks and infracomplete disks are still σ–disks and infracomplete disks, respectively. A LCS X is said to be infracomplete if the von Neumann bornology Mvon(X) has a basis consisting of infracomplete disks in X, or equivalently, σ–disks in X.

3 The construction and the commutativity of the triangle

Deﬁnition 3.1. 1. (“Operators”) A family A = {A(X,Y ): X,Y ∈ C} of algebras of operators associated to each pair of spaces X and Y in C is called an operator ideal if

OI1 A(X,Y ) is a nonzero vector subspace of L(X,Y ) for all X, Y in C; and

OI2 RTS ∈ A(X0,Y0) whenever R ∈ L(Y,Y0), T ∈ A(X,Y ) and S ∈ L(X0,X) for any X0, X, Y and Y0 in C. 2. (“Topologies”) A family P = {P(X): X ∈ C} of locally convex (Hausdorﬀ) topologies associated to each space X in C is called a generating topology if

0 GT1 σ(X,X ) ⊆ P(X) ⊆ Pori(X) for all X in C; and

GT2 L(X,Y ) ⊆ L(XP,YP) for all X and Y in C. 3. (“Bornologies”) A family M = {M(Y ): Y ∈ C} of convex vector (separated) bornologies associated to each space Y in C is called a generating bornology if

GB1 Mﬁn(Y ) ⊆ M(Y ) ⊆ Mvon(Y ) for all Y in C; and × M M GB2 L(X,Y ) ⊆ L (X ,Y ) for all X and Y in C.

Classical examples of these notions are the ideals Kp of precompact operators and P of absolutely summing operators (see e.g. [14]), the generating systems Ppc of precompact topologies (see e.g. [15]) and Ppn of prenuclear topologies (see e.g. [16, p. 90]), and the generating systems Mpc of precompact bornologies and Mpn of prenuclear bornologies (see e.g. [6]), respectively. The interesting fact among these examples is that “operators”, “topologies” and “bornologies” form vertices of a triangle and there are actions transforming one into another, which form edges of the triangle.

3 Deﬁnition 3.2. Let A be an operator ideal, P a generating topology and M a generating bornology on C.

1. (“Operators” → “Topologies”) For each X0 in C, the A–topology of X0, denoted by T(A)(X0), is the projective topology of X0 with respect to the family {T ∈ A(X0,Y ): Y ∈ C}. In other words, a seminorm p of X0 is T(A)(X0)–continuous if and only if there is a T in A(X0,Y ) for some Y in C and a continuous seminorm q of Y such that p(x) ≤ q(T x), ∀x ∈ X0. In this case, we call p an A–seminorm of X0.

2. (“Operators” → “Bornologies”) For each Y0 in C, the A–bornology of Y0, denoted by B(A)(Y0), is the inductive bornology of Y0 with respect to the family {T ∈ A(X,Y0): X ∈ C}. In other words, a subset B of Y0 is B(A)(Y0)–bounded if and only if there is a T in A(X,Y0) for some X in C and a topologically bounded subset A of X such that B ⊆ TA. In this case, we call B an A–bounded subset of Y0.

b 3. (“Topologies” → “Operators”) For X, Y in C, let O(P)(X,Y ) = L(XP,Y ) (resp. O (P)(X,Y ) = b L (XP,Y )) be the vector space of all continuous operators from X into Y which is still continuous with respect to the P(X)–topology (resp. which send a P(X)–neighborhood of zero to a bounded set). 4. (“Bornologies” → “Operators”) For X, Y in C, let O(M)(X,Y ) = L(X,Y ) ∩ L×(X,Y M) (resp. Ob(M)(X,Y ) = Lb(X,Y M)) be the vector space of all continuous operators from X into Y which send bounded sets to M(Y )–bounded sets (resp. which send a neighborhood of zero to an M(Y )–bounded set).

◦ ◦ 5. (“Topologies” “Bornologies”) For X, Y in C, the P (Y )–bornology of Y (resp. M (X)– topology of X) is deﬁned to be the bornology (resp. topology) polar to P(X) (resp. M(Y )). More precisely, a bounded subset A of Y is P◦(Y )–bounded if and only if the (absolute) polar

A◦ = {f ∈ Y 0 : |ha, fi| ≤ 1, ∀a ∈ A}

0 0 of A is a P(Yβ)–neighborhood of zero of the strong dual Yβ of Y . Similarly, a neighborhood V of ◦ ◦ 0 zero of X is a M (X)–neighborhood of zero if and only if V is M(Xβ)–bounded. Theorem 3.3 ([26]). Let A be an operator ideal, P a generating topology and M a generating bornology on C. 1. T(A) = {T(A)(X): X ∈ C} is a generating topology on C. 2. B(A) = {B(A)(Y ): Y ∈ C} is a generating bornology on C. 3. O(P) = {O(P)(X,Y ): X,Y ∈ C} is an operator ideal on C. 4. Ob(P) = {Ob(P)(X,Y ): X,Y ∈ C} is an operator ideal on C. 5. O(M) = {O(M)(X,Y ): X,Y ∈ C} is an operator ideal on C. 6. Ob(M) = {Ob(M)(X,Y ): X,Y ∈ C} is an operator ideal on C. 7. P◦ = {P◦(Y ): Y ∈ C} is a generating topology on C. 8. M◦ = {M◦(Y ): Y ∈ C} is a generating bornology on C. Let X and Y be LCS’s. J in L(X,Y ) is called a (topological) injection if J is one-to-one and relatively open. Q in L(X,Y ) is called a (topological) surjection if Q is open (and thus Q induces the topology of Y ). Q1 in L(X,Y ) is called a bornological surjection if Q1 is onto and induces the bornology of Y (i.e. for each bounded subset B of Y there is a bounded subset A of X such that Q1A = B). An operator ideal A on LCS’s is said to be injective, surjective or bornologically surjective if we can 1 induce T ∈ A(X,Y ) whenever T ∈ L(X,Y ) such that JT ∈ A(X,Y0), TQ ∈ A(X0,Y ) or TQ ∈ A(X1,Y ) 1 for some injection J in L(Y,Y0), surjection Q in L(X0,X) or bornological surjection Q in L(X1,X),

4 respectively. The injective hull Ainj, the surjective hull Asur and the bornologically surjective hull Absur of A is the intersection of all injective, surjective or bornologically surjective operator ideals containing A, respectively. Note that for operator ideals on Banach spaces, the notions of surjectivity and bornological surjectivity coincide. inj Associate to each normed space N the Banach space N = l∞(UN 0 ) and the injection JN in inj sur L(N,N ) defined by JN (x) = (hx, ai) . Similarly, we define N to be the normed space L1(UN ) = a∈UN0 {(λ ) ∈ ` (U ): P λ x converges in N} and Q : N sur → N to be the surjection defined by x 1 N x∈UN x N Q ((λ ) ) = P λ x. N x x∈UN x∈UN x Proposition 3.4 ([14, 3, 27]). 1. Let A be an operator ideal on Banach spaces.

inj inj A (E,F ) = {R ∈ L(E,F ): JF R ∈ A(E,F )}, sur sur A (E,F ) = {S ∈ L(E,F ): SQE ∈ A(E ,F )}.

2. Let A be an operator ideal on LCS’s. We can associate to each LCS Y a LCS Y ∞ and an injection ∞ ∞ 1 1 1 JY from Y into Y , and to each LCS X a LCS X and a bornological surjection QX from X onto X such that

inj ∞ ∞ A (X,Y ) = {R ∈ L(X,Y ): JY R ∈ A(X,Y )}, bsur 1 1 A (X,Y ) = {S ∈ L(X,Y ): SQX ∈ A(X,Y )}. In case N is a normed space, R ∈ L(X,N) and S ∈ L(N,Y ),

inj ∞ ∞ JN R ∈ A(X,N ) ⇐⇒ JN R ∈ A(X,N ), 1 1 sur SQN ∈ A(N ,Y ) ⇐⇒ SQN ∈ A(N ,Y ). The following result gives a full description of A–topologies and A–bornologies. Theorem 3.5 ([24]). Let A be an operator ideal on LCS’s. Let p be a continuous seminorm of a LCS −1 X and B an absolutely convex bounded subset of a LCS Y . Denote by Xp the normed space X/p (0) −1 equipped with norm kx + p (0)k = p(x), and by Y (B) the normed space ∪λ>0λB equipped with norm rB(x) = inf{λ > 0 : x ∈ λB}. Let Xep be the completion of Xp. Deﬁne Qp : X −→ Xp, Qep : X −→ Xep and JB : Y (B) −→ Y to be the canonical maps. We have

inj inj 1. p is an A–seminorm if and only if Qp ∈ A (X,Xp) if and only if Qep ∈ A (X, Xep). bsur 2. B is an A–bounded set if and only if JB ∈ A (Y (B),Y ). Whenever A is surjective, we can replace Absur by Asur. For operator ideals A on Banach spaces, Stephani [18, 20] achieved that O(T(A)) = Ainj and O(B(A)) = Asur. However, we have constructions O and Ob in the context of LCS’s. Unlike the Banach space version, they give rise to diﬀerent ideals. For example, let Mpc be the generating system of precompact b bornologies (i.e. the bornologies determined by totally bounded convex sets). Then Kp = O (Mpc) is the ideal of precompact operators (i.e. those sending a neighborhood of zero to a totally bounded set) and loc Kp = O(Mpc) is the ideal of locally precompact operators (i.e. those sending bounded sets to totally loc bounded sets). Randtke [15] indicated that Kp(X,Y ) = Kp (X,Y ) holds for all LCS Y if and only if X is a Schwartz space. On the other hand, it is straightforward to make the following observation. Proposition 3.6 ([26]). For a generating topology P and a generating bornology M on LCS’s, O(P) and Ob(P) give rise to the same ideal topology, namely

T(O(P)) = T(Ob(P)) = P,

and O(M) and Ob(M) give rise to the same ideal bornology, namely

B(O(M)) = B(Ob(M)) = M.

Moreover, O(P) and Ob(P) are injective, O(M) is bornologically surjective and Ob(M) is surjective.

5 In general, we have Proposition 3.7 ([26]). Let A be an operator ideal on LCS’s. We have 1. Ob(T(A)) ⊆ Ainj ⊆ O(T(A)). 2. Ob(B(A)) ⊆ Absur ⊆ O(B(A)). Let A be an operator ideal on LCS’s or Banach spaces. Adual denotes the operator ideal with compo- nents dual 0 0 0 A (X,Y ) = {T ∈ L(X,Y ): T ∈ A(Yβ,Xβ)}. We call A symmetric (resp. completely symmetric) if A ⊆ Adual (resp. A = Adual). Proposition 3.8 ([26]). (1) Let P be a generating topology on LCS’s. If the operator ideal A = O(P) is symmetric (resp. A = Ob(P) is symmetric) then P◦(Y ) = B(O(P))(Y ) (resp. B(Ob(P))(Y )) for all infrabarrelled LCS Y . (2) Let M be a generating bornology on LCS’s. If the operator ideal A = O(M) is symmetric (resp. A = Ob(M) is symmetric) then M◦(X) = T(O(M))(X) (resp. T(Ob(M))(X)) for all infrabarrelled LCS X. Deﬁnition 3.9. A generating topology P on LCS’s is said to have the subspace property if whenever Y is a subspace of a LCS X, YP is also a subspace of XP, i.e. the P–topology of Y coincides with the subspace topology inherited from the P–topology of X. See Jarchow [8] for a discussion on the case of generating topologies on Banach spaces. Proposition 3.10 ([26]). Let P be a generating topology on LCS’s and X be an infrabarrelled LCS. Then (a) Ob(P)dual(X,Y ) = Ob(P◦)(X,Y ), ∀LCS Y . (b) Ob(P◦)dual(X,Y ) ⊂ Ob(P)(X,Y ), ∀LCS Y . If, in addition, O(P) is symmetric or P has the subspace property then (b)0 Ob(P◦)dual(X,Y ) = Ob(P)(X,Y ), ∀LCS Y .

4 The operator theoretical, topological and bornological approaches in locally convex space theory

Let A be an operator ideal on LCS’s. Let X be a LCS and p a continuous seminorm of X. Denoted by −1 Xp and Xep the normed quotient space X/p (0) and its completion, respectively. We call p a Groth(A)- seminorm if there is a continuous seminorm q of X such that p ≤ q and the canonical map Qepq : Xeq −→ Xep belongs to A(Xeq, Xep). We call X a Groth(A)-space (resp. A-topological space) if all continuous seminorms of X are Groth(A)-seminorms (resp. A-seminorms). On the other hand, a bounded σ-disk B in a LCS Y is said to be Groth(A)-bounded if there is a bounded σ-disk A in Y such that B ⊆ A and the canonical map JAB : Y (B) −→ Y (A) belongs to A(Y (B),Y (A)). We call Y a co-Groth(A)-space (resp. A-bornological space) if all bounded σ-disks in Y are Groth(A)-bounded (resp. A-bounded). It is easy to see that a LCS X is a Groth(A)-space if and only if the topological completion Xe of X is a topological projective limit lim(Xeq, Qepq) of Banach spaces with linking maps from A (cf. [10]). ← Similarly, a LCS Y is a co-Groth(A)-space if and only if the bornological complete space Y equipped with the bornology determined by all of its bounded σ-disks is a bornological inductive limit lim(Y (B),JAB) → of Banach spaces with linking maps from A.

6 Theorem 4.1 ([26]). Let A be an operator ideal on LCS’s and X be a LCS. The following are all equivalent. (1) X is A–topological.

inj inj (2) For each continuous seminorm p on X, Qp ∈ A (X,Xp), or equivalently, Qep ∈ A (X, Xep). (3) Lb(X,Y ) ⊆ Ainj(X,Y ) for every LCS Y . (4) L(X,F ) = Ainj(X,F ) for every normed (or Banach) space F .

(5) idX ∈ L(XA,X), where XA is the LCS X equipped with the A–topology. Theorem 4.2 ([26]). Let A be an operator ideal on LCS’s and Y be a LCS. The following are all equivalent. (1) Y is A–bornological.

bsur (2) JB ∈ A (Y (B),Y ) for each bounded disk B in Y . (3) Lb(X,Y ) ⊆ Absur(X,Y ) for every LCS X. (4) L(N,Y ) = Absur(N,Y ) for every normed space N.

× A A (5) idY ∈ L (Y,Y ), where Y is the convex bornological vector space Y equipped with the A–bornology. In case Y is infracomplete they are all equivalent to (4)0 L(E,Y ) = Absur(E,Y ) for every Banach space E. If A is surjective we can replace Absur by A in all of the above statements. Theorem 4.3 ([26]). Let A be an operator ideal on LCS’s. Let G = T(A) be the ideal topology on LCS’s generated by A. A LCS X is A–topological if and only if

Lb(X,Y ) ∩ O(G)(X,Y ) = Ob(G)(X,Y )

for each LCS Y . Remark 4.4 ([26]). If we let M = B(A) then a LCS Y being A–bornological implies

Lb(X,Y ) ∩ O(M)(X,Y ) = Ob(M)(X,Y )

for each LCS X. We do not know if the converse is true.

Let A be an operator ideal on LCS’s. Denote by AB the operator ideal deﬁned on Banach spaces such that AB(E,F ) = A(E,F ) for every pair E and F of Banach spaces. Conversely, let A be an operator ideal on Banach spaces. There are many ways to extend A to an operator ideal A0 on LCS’s in the sense

that (A0)B = A. In [14], Pietsch mentioned six diﬀerent ways to extend A to an operator ideal on LCS’s. Among them, we are interested in

inf A = {RS0T ∈ L(X,Y ): T ∈ L(X,X0),S0 ∈ A(X0,Y0),R ∈ L(Y0,Y )}, rup A = {S ∈ L(X,Y ): ∀B ∈ L(Y,Y0), ∃A ∈ L(X,X0),S0 ∈ A(X0,Y0) such that BS = S0A}, lup A = {S ∈ L(X,Y ): ∀B ∈ L(X0,X), ∃A ∈ L(Y0,Y ),S0 ∈ A(X0,Y0) such that SB = AS0}, sup A = {S ∈ L(X,Y ): RST ∈ A(X0,Y0), for all T ∈ L(X0,X) and R ∈ L(Y,Y0)}.

Here, X,Y run through all LCS’s and X0,Y0 run through all Banach spaces. Theorem 4.5 ([24]). Let A be an operator ideal on Banach spaces. The Groth(Ainj)–topology coincides with the Arup–topology on every LCS, and the Groth(Asur)–bornology coincides with the Alup–bornology on every infracomplete LCS. In particular, we have

7 (a) A LCS X is a Groth(Ainj)–space if and only if X is an Arup–topological space. (b) An infracomplete LCS X is a co–Groth(Asur)–space if and only if X is an Alup–bornological space. (c) The A–topology (resp. A–bornology) coincides with the Groth(Ainj)–topology (resp. Groth(Asur)–bornology) on Banach spaces.

Proposition 4.6 ([26]). Let M be a generating bornology on LCS’s. (1) The Ob(M)–topology coincides with the Grothendieck topology generated by Ob(M) on every LCS.

(2) The O(M)–topology coincides with the (O(M)inj)sup–topology on each infracomplete LCS. B Proposition 4.7 ([26]). Let G be a generating topology on LCS’s. (1) The Ob(G)–bornology coincides with the Grothendieck bornology generated by Ob(G) on each infracomplete LCS. (2) The O(G)–bornology coincides with the (O(G)sur)sup–bornology on every infracomplete LCS. B Theorem 4.8 ([26]). (1) Let A be an operator ideal on LCS’s, and X a LCS. Then

dual 0 (a) X is A –bornological ⇒ Xβ is A–topological. If, in addition, X is infrabarrelled then

dual 0 (b) X is A –topological ⇒ Xβ is A–bornological.

0 dual (c) Xβ is A –bornological ⇒ X is A–topological. If, in addition to all above, A is also injective then

0 dual (d) Xβ is A –topological ⇒ X is A–bornological (2) Let A be a symmetric operator ideal on LCS’s, and X an infrabarrelled LCS. Then

0 (a) X is A–topological ⇔ Xβ is A–bornological. If, in addition, A is injective, then

0 (b) X is A–bornological ⇔ Xβ is A–topological. Theorem 4.9 ([26]). Let G be a generating topology on LCS’s and X be an infrabarrelled LCS. Then

b ◦ 0 b (a) X is O (G )–topological ⇒ Xβ is O (G)–bornological.

b ◦ 0 b (b) X is O (G )–bornological ⇒ Xβ is O (G)–topological.

0 b ◦ b (c) Xβ is O (G )–topological ⇒ X is O (G)–bornological.

0 b ◦ b (d) Xβ is O (G )–bornological ⇒ X is O (G)–topological. In case O(G) is symmetric or G has the subspace property, all above implications become equivalences.

8 5 Applications: Schwartz and Nuclear spaces

Deﬁnition 5.1 (see, e.g., [29, p. 14]). A continuous seminorm p on a LCS X is said to be precompact if 0 0 there exists a (λn) in c0 and an equicontinuous sequence {xn} in X such that

0 p(x) ≤ sup{|λnhx, xni| : n ≥ 1}, ∀x ∈ X.

Denote by Gpc(X) the locally convex (Hausdorff) topology on X defined by all precompact seminorms on X. It is easy to see that Gpc = {Gpc(X): X is a LCS} is a generating topology. It is a classical result (cf. [15] or [29]) that p is a precompact seminorm on a LCS X if and only if the canonical map Qp : X → Xp b b is precompact. Then, Kp = O(Gpc) is the ideal of all precompact operators, and Kp = O (T(Kp)) is the ideal of all quasi–Schwartz (i.e. precompact–bounded) operators between LCS’s. Definition 5.2. A LCS X is said to be a Schwartz space if every continuous seminorm p on X is precompact. We provide a new proof of the following classical result. Theorem 5.3 (see [29, pp. 17 and 26]). Let X be a LCS. The following are all equivalent. (a) X is a Schwartz space. (b) For each continuous seminorm p on X there is a continuous seminorm q on X such that p ≤ q and the canonical map Qpq belongs to Kp(Xq,Xp).

(c) Qp ∈ Kp(X,Xp) for every continuous seminorm p on X. (d) For any 0–neighborhood U in X there exists a 0–neighborhood V in X such that V ⊆ U and the canonical map from X0(U ◦) into X0(V ◦) is precompact.

(e) L(X,N) = Kp(X,N) for every normed (or Banach) space N.

b b (f) Kp(X,Y ) = L (X,Y ) for every LCS Y .

b (g) Kp(X,Y ) = Kp(X,Y ) for every LCS Y .

b (h) L(X,N) = Kp(X,N) for every normed (or Banach) space N.

(i) X is a Kp–topological space.

Proof. (a)⇔(c)⇔(e)⇔(i) are due to Theorem 4.1 and the injectivity of Kp. (a)⇔(b) because Kp = b O (Mpc) where Mpc is the generating bornology of precompact sets and Proposition 4.6 applies. (b)⇔(d) follows from the complete symmetry of the restriction (K ) of K to Banach spaces, i.e. (K )dual = (K ) . p B p p B p B (a)⇔(f)⇔(h) are consequences of Theorem 4.1. (i)⇒(g) is contained in Proposition 3.7 and Theorem 4.3. 0 Finally, for (g)⇒(h), denote by Nσ the LCS (N, σ(N,N )). For every T in L(X,N), T ∈ L(X,Nσ) = b b Kp(X,Nσ). Hence, by (g), T ∈ Kp(X,Nσ) = Kp(X,N) since N and Nσ carry the same (von Neumann) bornology.

0 Deﬁnition 5.4. A LCS Y is said to be a co–Schwartz space if its strong dual Yβ is a Schwartz space. Theorem 5.5. Let Y be a LCS. Consider the following statements. (a) Y is a co–Schwartz space. (b) For each bounded disk B in Y there is a bounded disk A in Y with B ⊆ A such that the canonical map JAB from Y (B) into Y (A) belongs to Kp(Y (B),Y (A)).

(d) L(N,Y ) = Kp(N,Y ) for every normed space N.

b (e) L (X,Y ) = Kp(X,Y ) for every LCS X.

9 (f) Y is a Kp–bornological space. We have (a)⇔(b)⇒(c)⇔ (d)⇔(e)⇔(f). Proof. (a)⇔(b) follows from the equivalence (a)⇔(b) in the last theorem and the complete symmetry of I (Kp)B. (c)⇔(d)⇔(e)⇔(f) are just examples of Theorem 4.2. (b)⇒(c) is trivial. Finally, the LCS K , where the index set I is uncountable, furnishes a counter–example of the missing implication. Deﬁnition 5.6. A continuous seminorm p on a LCS X is called an absolutely summing seminorm (= prenuclear seminorm in [29]) if there exists a σ(X0,X)–closed equicontinuous subset B of X0 and a positive Radon measure µ on B such that Z p(x) ≤ |hx, x0i|dµ(x0), ∀x ∈ X. B

Let Gas(X) be the locally convex (Hausdorﬀ) topology on X generated by the family of all absolutely summing seminorms on X. It is easy to see that the system Gas = {Gas(X): X a LCS} is a generating topology. A continuous operator T from a LCS X into a LCS Y is said to be absolutely summing if

T ∈ O(Gas)(X,Y ) = L(XGas ,Y ). In case X and Y are Banach spaces, T is absolutely summing if and only if T sends every weakly summable series in X to an absolutely summable series in Y . Denote by b b P = O(Gas) the injective ideal of all absolutely summing operators between LCS’s, and by P = O (Gas) the injective ideal of prenuclear–bounded operators [29]. A continuous operator T from a LCS X into a LCS Y is said to be nuclear if there exist a (λn) in 0 l1, an equicontinuous sequence {an} in X and a sequence {yn} contained in an infracomplete bounded disk B in Y such that T = Σnλnan ⊗ yn, i.e. T x = Σnλnan(x)yn for each x in X. Denote by N the ideal of all nuclear operators between LCS’s. Note that N is symmetric. It is more or less classical that P = Prup [29, p. 76], N = Ninf [29, p. 144] and P3 ⊂ N ⊂ P [29, p. 145] (in fact, we have B B B B B P2 ⊂ N , (Ninj)2 ⊂ N , cf. [13]). B B B B Deﬁnition 5.7. A LCS X is said to be nuclear if every continuous seminorm p on X is absolutely 0 summing. A LCS Y is said to be co–nuclear if its strong dual Yβ is nuclear. We provide a new proof of the following classical result. Theorem 5.8 (see [29, pp. 149 and 157]). Let X be a LCS. The following are all equivalent. (a) X is a nuclear space.

(b) Qp ∈ P(X,Xp) for every continuous seminorm p on X. (c) For each continuous seminorm p on X there exists a continuous seminorm q on X with p ≤ q such that the canonical map Qpq ∈ P(Xq,Xp).

(d) idX ∈ P(X,X). (e) P(X,Y ) = L(X,Y ) for every LCS Y .

(f) P(X,N) = L(X,N) for every normed space N.

(g) Qep ∈ N(X, Xep) for every continuous seminorm p on X. (h) N(X,F ) = L(X,F ) for every Banach space F . (i) For each continuous seminorm p on X there exists a continuous seminorm q on X with p ≤ q such that the canonical map Qepq ∈ N(Xeq, Xep). (j) For each 0–neighborhood V in X there is a 0–neighborhood U in X with U ⊆ V such that the canonical map X0(V ◦) → X0(U ◦) is nuclear. (k) Pb(X,Y ) = Lb(X,Y ) for every LCS Y .

10 (l) Lb(X,Y ) ⊆ P(X,Y ) for every LCS Y .

b (m) Kp(X,Y ) ⊆ P (X,Y ) for every LCS Y . (n) Lb(X,Y ) ∩ P(X,Y ) = Pb(X,Y ) for every LCS Y .

(o) X is a P–topological space. (p) X is a N–topological space. Proof. (a) ⇔ (b) ⇔ (d) ⇔ (e) ⇔ (f) ⇔ (l) ⇔ (o) ⇔ (p) are due to Theorem 4.1. Since P = Prup, we B have (a) ⇔ (c) by Theorem 4.5. (a) ⇔ (n) ⇔ (k) are due to Proposition 3.7 and Theorem 4.3. (k) ⇒ (m) is obvious. To prove (m) ⇒ (k) we employ the same trick as in Theorem 5.3. (c) ⇔ (i) follows from the fact that P3 ⊂ N ⊂ P. (i) ⇒ (g) ⇒ (h) are trivial. (h) ⇒ (l) because N ⊆ P and P is injective. (i)

⇒ (j) is ensured by the symmetry of NB. Finally, we prove (j) ⇒ (i). Let Vp = {x ∈ X : p(x) ≤ 1} be the 0–neighborhood associated to a continuous seminorm p on X. By (j), there is a continuous seminorm q on X such that Vq ⊆ Vp (i.e. p ≤ q) 0 0 ◦ 0 ◦ 00 0 ◦ 0 0 ◦ 0 and Qepq : X (Vp ) → X (Vq ) is nuclear. By the symmetry of NB, Qepq :(X (Vq )) → (X (Vp )) is nuclear, 00 0 ◦ 0 0 ◦ 0 00 too. Hence Qepq is absolutely summing. Now (X (Vq )) and (X (Vp )) are isometrically isomorphic to Xq 00 and Xp , respectively. By the injectivity of P, Qepq is absolutely summing. Repeating the same argument,

we shall have continuous seminorms q1 and q2 on X such that q ≤ q1 ≤ q2 and Qeqq1 and Qeq1q2 are both absolutely summing. Now p ≤ q and Q = Q Q Q ∈ P3 ⊆ N , and we are done. 2 epq2 epq eqq1 eq1q2 B B Remark 5.9. There are concepts of quasi–nuclear–seminorms, quasi–nuclear operators and quasi– nuclear–bounded operators, cf. [29]. They can be used to deﬁne nuclear spaces like P and N. However, they are simply, respectively, the N–seminorms, Ninj–operators and (T(Ninj))b–operators. Using the same kind of argument in Theorem 5.8, one can easily prepare a longer list of equivalences. We leave this to the interested readers. Theorem 5.10. Let Y be an infrabarrelled LCS. The following are all equivalent. (a) Y is a co–nuclear space.

(b) For each bounded disk B in Y there is a bounded disk A in Y with B ⊆ A such that the canonical map JAB from Y (B) into Y (A) is nuclear.

(e) Lb(X,Y ) ⊆ N(X,Y ) for every LCS X. (f) Y is an N–bornological space. Proof. Assume ﬁrst that Y is co–nuclear and B is a bounded disk in Y . Then B◦ is a 0–neighborhood in 0 00 ◦• 00 ◦• Yβ. Hence there is a bounded disk A in Y with B ⊆ A such that the canonical map Y (B ) → Y (A ) is absolutely summing. Since P is injective, the canonical map JAB from Y (B) into Y (A) is also absolutely summing. Do this twice more and we shall get (a) ⇒ (b) since P3 ⊆ N . (b) ⇒ (c), (d), (e) and each B B one of them ⇒ (f) are straightforward. We consider (f) ⇒ (a). Note that N is symmetric. Now Theorem 4.8(a) gives the desired conclusion. Proposition 5.11. Let X be an infrabarrelled LCS. Then X is a nuclear space (resp. co–nuclear space) 0 00 if and only if Xβ is a co–nuclear space (resp. nuclear space) if and only if Xββ is a nuclear space (resp. co–nuclear space).

Proof. In view of Theorem 4.9, it suﬃces to mention that the generating system Gas of absolutely summing 00 topology has the subspace property. As a result, an infrabarrelled LCS X is nuclear if and only if Xββ is nuclear. The other implications follow from this.

11 Since N ⊂ Kp we have the well–known Proposition 5.12. All nuclear (resp. co–nuclear) spaces are Schwartz (resp. co–Schwartz) spaces. Finally, the readers are referred to [26] for the full exploration of results surveyed in this paper.

References

[1] J. Conradie and G. West, “Topological and bornological characterisations of ideals in von Neumann algebras: II”, Integral equations and operator theory, to appear. [2] A. Defant and K. Floret, “Tensor norms and operator ideals”, Math. Studies 176, North–Holland, Amsterdam, 1993. [3] L. Franco and C. Piñeiro,“The injective hull of an operator ideal on locally convex spaces”, Manuscripa Math., 38, 1982, 333–341. [4] A. Grothendieck, “Rèsumeéde la theóriemmmétrique des produits tensoriels topologiques”, Bol. Soc. Mat. SãoPaulo, 8, 1956, 1–79. [5] H. Hogbe-Nlend, “Bornologies and functional analysis”, Math. Studies 26, North–Holland, Amster- dam, 1977. [6] H. Hogbe-Nlend, “Nuclear and co–nuclear spaces”, Math. Studies 52, North–Holland, Amsterdam, 1981.

[7] G. J. O. Jameson, “Summing and nuclear norms in Banach space theory”, London Math. Soc. Students Text Vol. 8, Cambridge University Press, Cambridge, 1987. [8] H. Jarchow, “Locally Convex Spaces”, Teubner, Stuttgart, 1981. [9] H. Jarchow, “On certain locally convex topologies on Banach spaces”, Functional Analysis: Surveys and Recent Results, III, K. D. Bierstedt and B. Fuchssteiner (eds.), North–Holland, Amsterdam, 1984, 79–93. [10] H. Junek, “Locally convex spaces and operator ideals”, Teubner–texte zur Math., Band 56, Teubner, Leipzig, 1983.

[11] M. Lindstorm, “A characterization of Schwartz spaces”, Math. Z. 198, 1988, 423–430. [12] P. Michor, “Functors and categories of Banach spaces”, Lecture Notes in Mathematics 651, Springer- Verlag, Berlin-Heidelberg-New York, 1978. [13] A. Pietsch, “Nuclear Locally Convex Spaces”, Springer–Verlag, Berlin–Heidelberg–New York, 1972.

[14] A. Pietsch, “Operator Ideals”, North–Holland, Amsterdam, 1980. [15] D. Randtke, “Characterizations of precompact maps, Schwartz spaces and nuclear spaces”, Trans. Amer. Math. Soc. 165, 1972, 87–101. [16] H. H. Schaefer, “Topological Vector Spaces”, Springer–Verlag, Berlin–Heidelberg–New York, 1971.

[17] R. Schatten, “Norm ideals of completely continuous operators”, Ergebn. Math. Grenzgeb. 27, Springer, 1960. [18] I. Stephani, “Injektive operatorenideale über der gesamtheit aller Banachräumeund ihre topologische erzeugung”, Studia Math. 38, 1970, 105–124. [19] I. Stephani, “Surjektive operatorenideale über der gesamtheit aller Banachräume”, Wiss. Z. FSU Jena, Math–Nat. R. Jg. vol. 21, 1972, 187–216.

12 [20] I. Stephani, “Surjektive operatorenideale über der gesamtheit aller Banachräumeund ihre Erzeu- gung”, BeiträgeAnalysis, vol. 5, 1973, 75–89. [21] I. Stephani, “Generating system of sets and quotients of surjective operator ideals”, Math. Nachr. 99, 1980, 13–27.

[22] I. Stephani, “Generating topologies and quotients of injective operator ideals”, in “Banach Space Theory and Its Application (proceedings, Bucharest 1981)”, Lecture Notes in Math., vol. 991, Springer–Verlag, Berlin–Heidelberg–New York, 1983, 239–255. [23] G. West, “Topological and bornological characterisations of ideals in von Neumann algebras: I”, Integral equations and operator theory, to appear.

[24] Ngai-Ching Wong, “Topologies and bornologies determined by operator ideals, II”, Studia Math. 111 (2), 1994, 153–162. [25] Ngai-Ching Wong, “Yau-Chuen Wong”, Southeast Asian Bull. Math., vol. 20, no. 1, 1996, 1–3. [26] Ngai-Ching Wong, “Triangle of operators, topology, bornology”, in the Proceedings of the 3rd Inter- national Congress of Chinese Mathematicans (2004), 395-421, AMS/IP, Studies in Advanced Math. 42, 2008. [27] Ngai-Ching Wong and Yau-Chuen Wong, “The bornologically surjective hull of an operator ideal on locally convex spaces”, Math. Narch. 160, 1993, 265–275. [28] Yau-Chuen Wong, “The topology of uniform convergence on order–bounded sets”, Lecture Notes in Mathematics, vol. 531, Springer–Verlag, Berlin–Heidelberg–New York, 1976. [29] Yau-Chuen Wong, “Schwartz spaces, nuclear spaces and tensor products”, Lecture Notes in Mathe- matics, vol. 726, Springer–Verlag, Berlin–Heidelberg–New York, 1979. [30] Yau-Chuen Wong and Ngai-Ching Wong, “Topologies and bornologies determined by operator ideals”, Math. Ann. 282, 1988, 587–614. [31] Yau-Chuen Wong, “Introductory theory of topological vector spaces”, Pure and Applied Mathemat- ics, vol. 167, Marcel Dekker, Inc., New York–Basel–Hong Kong, 1992.