Mackey +0-Barrelled Spaces Stephen A

Home , Barrelled set, Barrelled space, Mackey space, Mackey topology

Advances in Mathematics 145, 230238 (1999) Article ID aima.1998.1815, available online at http:ÂÂwww.idealibrary.com on

Mackey +0-Barrelled Spaces Stephen A. Saxon*

Department of Mathematics, University of Florida, P.O. Box 118105, Gainesville, Florida 32611-8105 E-mail: saxonÄmath.ufl.edu

and

Ian Tweddle*

Department of Mathematics, University of Strathclyde, Glasgow G11XH, Scotland, United Kingdom E-mail: i.tweddleÄstrath.ac.uk CORE Metadata, citation and similar papers at core.ac.uk

Provided by Elsevier - PublisherReceived Connector March 25, 1998; accepted December 14, 1998

In the context of ``Reinventing weak barrelledness,'' the best possible versions of the RobertsonSaxonRobertson Splitting Theorem and the SaxonTweddle Fit and Flat Components Theorem are obtained by weakening the hypothesis from

``barrelled'' to ``Mackey and +0-barreled.'' An example showing that the latter does not imply the former validates novelty, answers an old question, and completes a robust linear picture of ``Mackey weak barrelledness'' begun several decades ago. 1999 Academic Press Key Words: weak barrelledness; Mackey topology; splitting theorem.

1. INTRODUCTION

Barrelled spaces have occupied an important place in the theory of locally convex spaces since its earliest days. This is probably due to the fact that they provide a vehicle for generalizing certain important properties of Banach spaces; for example, they form the class of domain spaces for a natural generalization of Banach's closed graph theorem and, in their dual characterization, they embody the conclusion of the BanachSteinhaus theorem for continuous linear functionals. Unlike Banach spaces, they are closed under the taking of inductive limits, countable-codimensional subspaces, products, etc. A barrel in a locally convex space is a closed

* We are grateful for support and hospitality from EPSRC GRÂL67257 and Strathclyde University, and for helpful discussions with Professors J. K. Brooks, J. Diestel, S. McCullough, P. L. Robinson, and L. M. Sanchez Ruiz. 230 0001-8708Â99 30.00 Copyright 1999 by Academic Press All rights of reproduction in any form reserved. MACKEY +0 -BARRELLED SPACES 231 absorbing balanced convex set; every locally convex space has a base of neighborhoods of the origin made up of barrels and barrelled spaces are precisely those locally convex spaces with the property that each of their barrels is a neighborhood of the origin. The term barrel is just the translation of the Bourbaki term ``tonneau,'' which has likewise been adopted in direct translation in other languages (German, Spanish, Russian). The dual characterization of barrelledness is that each pointwise bounded set of continuous linear functionals is equicontinuous. Recent decades have seen the systematic study of spaces that are somewhat more than barrelled (strong barrelledness), or somewhat less (weak barrelledness (WB)), a dichotomy emphasized in Chapters 8 and 9 of Perez Carreras and Bonet [11]. The AmemiyaKoÃmura [1] and Banach Mackey theorems, combining both areas, prove that, under metrizability, all the strongÂweak barrelledness properties between (inclusively) Baire- likeness and dual local completeness (dlc) coincide, even though generally, barrelledness is properly between Baire-likeness and the much weaker dlc (cf. [15, 5, 22]). Further improvement in either direction seems impossible. In the WB direction, a locally convex space is inductive if it is the inductive limit of each increasing covering sequence of subspaces, and is primitive if and only if it becomes inductive when endowed with the Mackey topology. Recall that metrizable locally convex spaces, like barrelled spaces, already have the Mackey topology, the finest locally convex topology that reproduces the given dual. Primitivity is the very weakest in the panoply of WB properties [24] and coincides with inductivity but not dlc under metrizability, leaving metrizable WB with precisely two distinct classes of spaces [6, 25]. Mackey WB, on the other hand, has seven such classes, as was just proved in [23] with the aid of the present paper. The proof of [23] features the following answers to questions posed in 1991 and 1971: (i) The WB properties between C-barrelled and dlc are equivalent under the Mackey topology, and (ii) there exists a Mackey space with property (C) which is not l -barrelled. The popular notion of l -barrelledness, also known as _-barrelledness or |-barrelledness [2, 9, 11], abstracts the relaxed need to assert that a pointwise bounded sequence of continuous linear functionals is equicontinuous. Rather more sophisticated is Husain's idea [7] of countable barrelledness (here called +0-barrelledness, as in [11]). Grothendieck, in 1954, had already introduced and shown the importance of (DF )-spaces, a class of spaces which abstract certain properties of the strong dual of a Frechet space: a locally convex space is a (DF )- space if it has a fundamental sequence of bounded sets and each strongly bounded subset of the dual space which is the union of a sequence of equicontinuous sets is itself equicontinuous. Husain's idea was simply to require that the union of any sequence of equicontinuous sets which is pointwise 232 SAXON AND TWEDDLE bounded be equicontinuous. It is the strongest WB property distinct from barrelledness. Many distinguishing examples have been found; in particular Morris and Wulbert [10] pointed out that C[0, 0) with the topology of uniform convergence on the compact subset of [0, 0)is+0-barrelled but not barrelled; here 0 denotes the first uncountable ordinal. Soon after Husain made his definition, it was realized that underlying properties often elevate +0-barrelledness to barrelledness; for example, any seperable - +0 barrelled space is barrelled, as is any metrizable, or bornological, - +0 barrelled space. One such consideration, posed in [8, p. 63], appears to have remained open: Can an +0-barrelled space be a Mackey space without being barrelled? In this connection we note that Morris and Wulbert specifically proved that their space is not a Mackey space. One of the achievements of the present work is to provide an example of a Mackey

+0-barrelled space which is not barrelled. Another is to illustrate how this newly completed structural picture of Mackey WB [23] may foster optimal versions (below) of established theorems. In the last two decades there has been much interest in the barrelled countable enlargement problem: Given a (Hausdorff ) barrelled space which does not have its finest locally convex topology, is it possible to extend the dual space by +0 dimensions in such a way that the space is barrelled in the Mackey topology determined by the enlarged dual? Ideas associated with this problem are to be found in Amemiya and KoÃmura [1] and De Wilde and Tsirulnikov [3], but it was probably Wendy Robertson who first stated the problem explicitly [14]. The general problem appears to be still unresolved, although positive answers are known for a large number of special cases, including all infinite-dimensional metrizable barrelled spaces [16, 21]. For a survey of the state of knowledge up to 1994 see [29]. One approach to constructing a barrelled countable enlargement for a given barrelled space is to look for a dense (barrelled) subspace whose codimension is equal to the dimension of the spacethe space is then said to be (barrelledly) fit. Pursuing this aim, W. Robertson, S. A. Saxon, and A. P. Robertson established their Splitting Theorem [12]: Given a Hamel basis in a Hausdorff barrelled space E, the space can be expressed as the topological direct sum of subspaces EC and ED , where EC (ED) is the span of the basis vectors with continuous (discontinuous) coefficient functionals; moreover, EC has its finest locally convex topology. Subsequently, Saxon and Tweddle [27] showed (under an assumption weaker than the

Generalized Continuum Hypothesis) that ED is fit. In fact, they showed that if E is merely primitive and the algebraic split is still topological, then

ED is still fit when infinite-dimensional. However, simple examples (below) show that primitivity is not, nor is even +0-barrelledness, sufficient to ensure either a topological split or the finest locally convex topology on

EC . And if E is Mackey and has all the WB properties except barrelledness MACKEY +0 -BARRELLED SPACES 233

and +0-barrelledness, both conclusions of the Splitting Theorem may yet fail. But if E is Mackey and +0-barrelled, it is shown below that the conclusions hold. The Splitting and the Fit and Flat Components Theorems, then, have their best forms in the properly wider setting of Mackey - +0 barrelled spaces. A space E (Hausdorff, locally convex with real or complex scalar field) is +0-barrelled (l -barrelled ) if each pointwise bounded sequence of continuous seminorms (linear forms) is equicontinuous (cf. [11, 24]). O O O Obviously, barrelled +0-barrelled l -barrelled, and barrelled Mackey. The Mackey WB relational picture is linear, and any Mackey - l barrelled space has all the WB properties except possibly +0-barrelledness and barrelledness [24].

2. EXAMPLE

In the remainder of the discussion let 1 denote an uncountable set, and for each S/1 canonically identify l (S) as a subspace of the Banach space l (1 ), with l (<)=[0]. We will say that a member : of (l (1 ))$ has countable support if there is a countable S / 1 such that :(l (1"S)) =[0]. The unit ball of the non-reflexive Banach space l 1(1 )isnot _(l 1(1 ), l (1 ))-compact, as the following lemma also implies, for its members have countable support.

Lemma 2.1. If each member of a _((l (1 ))$, l (1 ))-compact set C has countable support, then there is a countable S/1 such that all members of C vanish on l (1"S). Proof. Suppose not. The first uncountable ordinal 0 is a well ordered set whose initial segments are countable. An easy transfinite induction on

0 yields a family [f: , S: , u:]: # 0 such that

1. [f:]: # 0/C;

2. [S:]: # 0 is a family of countable pairwise disjoint subsets of 1; 3. each u: # l (S:) with &u:&=1 and f: (u:){0; 4. f: (l (1";: S;))=[0].

Indeed, given # # 0 and well chosen f: , S: , u: for all :<#, the supposition and hypothesis guarantee f# # C which vanishes, not on all of l (1":<# S:), but on all of l (1"T) for some countable T/1. Thus f# is nonzero at some u# # l (T":<# S:) of unit norm, and one takes

S#=T":<# S:. 234 SAXON AND TWEDDLE

Some 0n=[: # 0:|f: (u:)|>1Ân] is uncountable, and compactness gives f # C such that each _ ((l (1 ))$, l (1 ))-neighborhood of f contains uncountably many elements of [f:]: # 0n . There is a countable Q / 1 such that f vanishes on l (1"Q), and since 0n has uncountable cofinality, there exists some :0 # 0n such that Q and S: are disjoint for each : # A=

[: # 0n : :>:0]. Since A is well-ordered, we may inductively define u # l (: # A S:) so that if : # A and S= [S:$ : :$#A, :$<:],then

+u: if | f: (u | S+u:)|>1Ân, u | S:= {&u: otherwise.

One interprets u | S as a member of both l (S) and l (1 ). Note that

&u&=1. Preserving notation, (4) implies | f: (u)|=| f:(u| S+u | S:)|= | f: (u | S\u:)|>1Ân since, for any complex numbers a and b, either

|a+b||b|or|a&b||b|. Now Q and : # A S: are disjoint, so f(u)=0, and thus |( f:& f )(u)|>1Ân for all : # A. But this means that f+[nu]% misses [f: ]: # A , hence intersects [f:]: # 0n in a (countable) subset of K [f: ]::0 , a contradiction.

Example 2.1. For each non-empty countable subset S/1 and for each u # l (1 ), define pS (u)=supx # S |u(x)| and give E=l (1 ) the topology generated by the seminorms pS . Then E is a Mackey +0-barrelled space that is not barrelled. Proof.IfC is any _(E$, E)-compact set, the lemma provides a countable S such that l (1"S)/C =, and in the Banach subspace l (S) the barrel C% & l (S) is a 0-neighbourhood. Thus C% is a 0-neighborhood in E,andE is Mackey. Any countable union S of countable sets is countable, and the Banach subspace l (S)is+0 -barrelled; thus so is E. The unit ball of the Banach space l (1 ) is a barrel that is not a 0-neighborhood in E, so E is not barrelled. K

3. BETTER THEOREMS

All the barrelled hypotheses of [12, 27] relax to [Mackey7 +0-barrelled], as the following pivotal case illustrates.

Theorem 3.1 (Splitting Theorem). Let E be a Mackey +0-barrelled space and let [xi , fi ]i # B be a biorthogonal system in E_E* whose first coordinates form a Hamel basis in E. If

EC =sp([xi : i # B, fi # E$]) and ED=sp([xi : i # B, fi Â E$ ]) MACKEY +0 -BARRELLED SPACES 235

then E is the topological direct sum of EC and ED , and EC has its strongest locally convex topology. Proof. Since E is Mackey, we only need show that E$ accommodates the conclusion; i.e., we only need show that if g # E* vanishes on ED ,then g # E$. The Tychonoff theorem and a routine argument show that in the _(E*, E)-topology,

K= : *i g(xi ) fi : I/B is countable and : |*i |1 {i # I i # I = is a closed subset of a compact set, hence is compact. But each

i # I *i g(xi ) fi # E$, since the countable _ (E$, E)-bounded set [g(xi) fi : i # I] is necessarily equicontinuous. Hence K is _(E$, E )-compact. Therefore the balanced convex K is equicontinuous on the Mackey space E, and each

Kn= : g(xi) fi : In consists of n elements of B /nK {i # I = n is also equicontinuous. But n Kn is _(E$, E )-bounded, so the +0-barrel n Kn%=(n Kn)% is a 0-neighborhood on which g is nummerically bounded (by 1), proving g # E$. K The hypothesis cannot be weakened within the WB context. When E is - non S_ , then EC must be finite-dimensional and the result obviously holds, but ``non-S_'' is neither weaker nor stronger than ``barrelled'' [24].

Example 3.1. Let E be an +1-dimensional vector space with Hamel basis A. The topology T generated by the seminorms which vanish on all but countably many elements of A is clearly +0 -barrelled, and the finer compatible Mackey topology {(E, E$) preserves l -barrelledness, so it has all the WB properties other than +0-barrelledness and barrelledness [23]. Select y # A and put B=[x+y: x # A]. With either topology the Hamel basis B determines ED=sp[(y )]is1-dimensional and EC is dense, so the algebraic splitting is not topological, and EC nonclosed means it is not complete, hence does not have its finest locally convex topology. That is, both conclusions of the above theorem may simultaneously fail if the hypothesis omits ``Mackey'' or if ``+0-barrelled'' is replaced by any combination of strictly weaker properties within the purview of [24].

We may now speak of the fit and flat components of Mackey +0-barrelled spaces E, since [27, Theorem 4] ED is fit if it is merely primitive, the weakest of all the properties found in [24]. Recall that a space E is (barrelledly) fit [18] if it has a dense (barrelled) subspace with (the largest possible) codimension=dim(E), and is flat (the extreme opposite of 236 SAXON AND TWEDDLE fit) if it has no proper dense subspaces at all; i.e., if E$=E*. Since components of Mackey spaces are Mackey, they are flat if and only if they have their strongest locally convex topology. It is apparently still open as to whether every barrelled, fit space is barrelledly fit [18]. A positive answer would give the same for the barrelled countable enlargement question under assumption of the Continuum Hypothesis (CH): If E is barrelled and non-flat, so is the component ED , which is thus uncountable-dimensional

[17, Corollary B], and if the barrelled, fit ED is therefore barrelledly fit, then E has a dense barrelled subspace of codimension at least c, which implies that E has a barrelled countable enlargement [13, Theorem 5].

The existing proof that ED is fit posits a condition on +, an arbitrary infinite cardinal [27]: Condition (2). Given any set A of cardinality less than +, there exists a set F of functions f form A into the set N of natural numbers such that the cardinality of F is at most + and such that for any g: A Ä N there exists f # F such that f(a)>g(a) for all a # A.

Condition (2) obviously holds for +=+0 , and also in general under the assumption of the generalized CH whereby the set F of all functions Ä |A| +0 |A| |A| g: A N would have cardinality +0 (2 ) =2 + for |A|<+. Thus the general assumption of Condition (2) is ZFC-consistent. For the special case + =+1, Condition (2) simply requires a dominating set F of func- Ä tions f: N N with |F|+1 ; essentially by definition (see penultimate paragraph, p. 138 of [19]), the least cardinality of such an F is the dominating cardinal d (+1). Therefore the assumption that +1 satisfies Condition (2) is equivalent to saying d=+1 , and is strictly weaker

[4, Sect. 3] than assuming c=+1 (the CH). Two concluding illustrations use the Splitting Theorem and set-theoretic assumptions precisely as in the original barrelled space versions [27]. The second is more than just a corollary of the first because of its significantly weaker assumption.

Theorem 3.2 (Assume Condition (2) generally). Every Mackey - +0 barrelled space E splits with respect to a given basis into a fit component

ED and a flat component EC .

Theorem 3.3 (Assume +1=d). Every non-flat Mackey +0-barrelled space E has a dense subspace of codimension at least +1 . Does the last theorem admit a more relaxed WB hypothesis? This time, E of Example 3.1 is silent, since it is fit [26, Example 4.1]. Note that if a non-flat E is merely primitive, there exists a dense +0-codimensional subspace F under the assumption that dim(E) is a non-measurable cardinal MACKEY +0 -BARRELLED SPACES 237

[24], and F is barrelled if E is [17]. But if E contains a dense (+0-) barrelled subspace of codimension at least b, then E has an (+0-) barrelled countable enlargement [20, 26, 28]. Because the bounding cardinal b always satisfies +1bdc, Theorem 3.3 and its assumption that +1=d may be particularly appropriate; property preserving countable enlargements exist for the spaces of Example 3.1 if and only if one assusmes +1=b [26].

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