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Acta Universitatis Carolinae. Mathematica et Physica

Warren B. Moors; Sivajah Somasundaram

Some recent results concerning weak Asplund spaces

Acta Universitatis Carolinae. Mathematica et Physica, Vol. 43 (2002), No. 2, 67--86 Persistent URL:http://dml.cz/dmlcz/702085

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2002 ACTA UNIVERSITATIS CAROLINAE - MATHEMATICA ET PHYSICA VOL. 43, NO. 2

Some Recent Results Concerning Weak Asplund Spaces

WARREN B. MOORS and SIVAJAH SOMASUNDARAM Hamilton

Received 14. March 2002

This paper provides a gentle introduction to the study of weak Asplund spaces. It begins with a brief historical review of the development of weak Asplund spaces; starting from the earliest results concerning the differentiability of convex functions on U through to the most recent developments concerning possible characterization of these spaces.

Along the way the classes of Gateaux differentiability spaces, Stegall spaces and fragmentable spaces are introduced and their relationship with weak Aslund spaces reviewed. Following this, we summarize some of the most recent attempts at distinguish- ing the classes of weak Asplund spaces; Stegall spaces and fragmentable spaces. We conclude the paper by examining a class of topological spaces, namely the class of weakly Stegall spaces, that may be useful in the problem of distinguishing the class of weak Asplund spaces from the class of Gateaux differentiability spaces.

1. Introduction

This paper has been designed to be read by someone who is presently unfamiliar with the study of weak Asplund spaces, with the aim of leading them through the subject and bringing them to the forefront in several areas of the subject. To facilitate this the paper has been divided into three sections.

In the first section we give a brief historical account of the development of weak Asplund spaces, then in section two we present some of the most recent attempts at distinguishing the various classes of Banach spaces that have been associated with Asplund spaces. Finally, in section three we develop the theory of weakly

Department of Mathematics, The University of Waikato, Private Bag 3105, Hamilton, New Zealand e-mail: moors@math. waikato.ac.nz; e-mail: [email protected]

AMS2 (2000) subject classification- 54C60, 26E25, 54C10.

Key words: weak Asplund, Stegall space, weakly Stegall space, fragmentable space.

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Stegall spaces and explore how these spaces are related to the study of weak Asplund spaces.

A real-valued function cp defined on a non-empty open convex subset A of a normed linear space X is said to be convex if,

cp(Xx + (1 — X) y) < Xcp(x) + (1 — X) cp(y) for all x,ye A and X e [0,1] . We say that cp is Gateaux differentiable at a point x e A if there exists a continuous linear functional x* e X* such that

x*(y) = l i m ^X + Xy} ~ *to for a l l y e X .

In this case, the linear functional x* is called the Gateaux derivative of cp at x. If the limit above is approached uniformly with respect to all y e 5x-the closed unit ball of X, then cp is said to be Frechet differentiable at the point xe A and x* is called the Frechet derivative of cp at x. The differentiability properties of convex functions on Banach spaces have been studied for many years. The first result concerning the differentiability of convex functions is the classical result that every continuous convex function defined on IR is differentiable everywhere except for (at most) a countable set (see [30, p. 9]). This result generalizes to IRn (either by using Fubini's theorem or the Kuratowski-Ulam theorem) except that one must replace "countable set" by either "measure zero set" or "first category set" (see [30, p. 11]). Indeed even the continuous convex function (x, y) i—• |x| is non-differentiable on an uncountable set. The question of how to extend these results to infinite dimensional spaces has several difficulties. For instance, what does one mean by "except for a small set" in infinite dimensions? The two natural candidates are null sets (in a measure theoretic sense) and first category sets (in a Baire categorical sense). The first notion leads to difficulties as there is no natural notion of Haar-measure on an infinite dimensional Banach space (as it is not locally compact). However, the other notion of "smallness" readily lends itself to infinite dimensions. Indeed, using this approach S. Mazur, [24] showed in 1933 that every continuous convex function defined on a separable Banach space is Gateaux differentiable everywhere except for (at most) a first category set. This result was revisited three decades later by J. Lindenstrauss, [23] who showed in 1963 that for reflexive separable Banach spaces the conclusions of Mazur's theorem may be strengthened to: Frechet differentiable everywhere except for (at most) a first category set.

Five years later (i.e., in 1968) E. Asplund introduced two classes of Banach spaces defined according to the differentiability properties of the continuous convex functions defined on them (see [1]). He called the Banach spaces on which every continuous convex function defined on a non-empty open convex subset is Gateaux differentiable everywhere except at the points of a first category set, weak differentiability spaces while he called the Banach spaces on which every 68

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continuous convex function defined on a non-empty open convex subset is Frechet differentiable everywhere except on a first category set strong differentiability spaces. These spaces have subsequently become known as weak Asplund spaces (see [22]) and Asplund spaces respectively (see [27]). In 1968 Asplund showed that every Banach space that can be renormed to have an equivalent rotund (equivalent locally uniformly rotund) dual norm is a weak differentiability (strong differen- tiability) space (see [1]). This result is an improvement upon Mazur's (Linden- Strauss') result since every separable (separable reflexive) Banach space space can be renormed to have an equivalent rotund (equivalent locally uniformly rotund) dual norm.

Since the time of Asplund a considerable volume of literature has been written on Asplund spaces and it was one of the major achievements of functional analysis in the late 70's (in fact, in 1978) when the class of Asplund spaces was characterized as those Banach spaces whose dual spaces possess the Ra- don-Nikodym property. This was the culmination of the results from the papers [27], [33] and [34]. Subsequent to this many other characterizations as Asplund spaces have been discovered (see for example [4], [7], [10] and [30]).

By contrast, our knowledge of weak Asplund spaces is rather thin. However, several partial results are known and are summarized in [8],

In an attempt to characterize the class of weak Asplund spaces Larmen and Phelps (in 1979) considered the following class of Banach spaces (see [22]).

A Banach space X is called a Gateaux differentiability space (or GDS for short) if each continuous convex function defined on a non-empty open convex subset A or X is Gateaux differentiable at the points of a dense (but not necessarily a Gs) subset of A. Therefore, every weak Asplund space is a Gateaux differentiability space. However, the status of the reverse implication remains unresolved. The importance of Gateaux differentiability spaces (apart from their obvious similarily to weak Asplund spaces) stems from the fact that they admit a dual characterization analogous to that of Asplund spaces. The proof of this characterization was done in two parts. The first, in 1979 by Larmen and Phelps (see [22]) and the second by M. Fabian in 1988 (unpublished).

Theorem 1 [30, Proposition 6.5] A Banach space X is a Gateaux differen- tiability space if, and only if, every non-empty weak* compact convex subset of X*

is the weak* closed convex hull of its weak* exposed points.

Recall that a point x* in a weak* compact convex subset C of X* is weak*

exposed there exists an element x e X such that x*(x) > y*(x) for all y* e C\ {**}.

That is, the weak* continuous linear functional x: X* -> IR defined by, x(y*) := y*(x) attains its maximum value on C at the single point x*.

In another attempt to characterize the class of weak Asplund spaces C. Stegall (in 1983) introduced the following class of topological spaces, which are defined in terms of minimal uscos (see [35]). A set-valued mapping cp : X -» 2Y acting

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between topological spaces X and Y is called an usco mapping if for each x e X, cp(x) is a non-empty compact subset of Y and for each open set W in Y, {xe X :

<p(x) ^ IV} is open in X. An usco mapping cp : X -> 2Y is called a minimal usco if its graph does not properly contain the graph of any other usco defined on X.

A topological space X is said to belong to the class of Stegall spaces if for every Baire space B and minimal usco cp : B -> 2X, cp is single-valued at the points of a residual subset of B. Correspondingly, we say that a Banach space X belongs to class(^) if (X*, weak*) belongs to the class of Stegall spaces.

The relationship between Stegall spaces and weak Asplund spaces is established through the subdifferential mapping. Let q>:A->R be a continuous convex function defined on a non-empty open convex subset A of a Banach space X. Then the subdifferential mapping of (p is the mapping dcp : A -> 2X* defined by,

dcp(x): = {x* e X* : x*(y — x) < cp(y) — cp(x) for all ye A}.

It is well-known that the subdifferential mapping is a norm-to-weak* usco (see [30, p. 19]). To establish the relationship between Stegall spaces and weak Asplund spaces we need one more fact.

Lemma 1 [5]. Let cp : X -> 2Y be an usco mapping acting between topological spaces X and Y. Then there exists a minimal usco mapping i/t: K -> 2y such that

\j/(x) <= cp(x) for all xe X (i.e., every usco mapping contains a minimal usco mapping).

Theorem 2 [35]. Every member of c\ass(9)) is weak Asplund.

Proof. Suppose X e class(^). Let A -> IR be a continuous convex function defined on a non-empty open convex subset A of X and let \jj : A -> 2X* be a minimal usco on A such that \l/(x) ^ dcp(x) for all xe A. Since A is a Baire space and (X*, weak*) belongs to the class of Stegall spaces there exists a residual subset R of A on which \// is single-valued. Now let o : A -> (X*, weak*) be any selection of \j/ and let x e i?. We claim that cp is Gateaux differentiable at x. For any y e X and A > 0 we have, by the definition of dcp, that

,(x)(>)<<*

X +

y-^<,(

X +

Ay)(y).

Since <r is norm-to-weak* continuous at x we have that,

a{X){y)=lim(p{x + ^-<p{x).

This means that cp is Gateaux diffentiable at x with derivative <T(X); which

completes the proof. • To date, the class(^) spaces are the largest known well-behaved subclass of

weak Asplund spaces.

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A further class of topological spaces that has played a role in the study of weak Asplund spaces is the class of fragmentable spaces. The notion of fragmentability was introduced by Jayne and Rogers in 1985 (see [11]) in regard to selection theorems. However, this class of spaces also has implications for the study of weak Asplund spaces. A topological space X is said to be fragmentable if there exists a metric d on X such that for every s > 0 and every non-empty set A of X there exists a non-empty subset B of A that is relatively open in A and d-diam(B) < s.

The corresponding notion for Banach spaces is the following: A Banach space X belongs to & if (X*, weak*) is fragmentable.

To establish the connection between fragmentability and Stegall spaces we need the following characterization.

Lemma 2 [8]. Let cp : X -» 2Y be an usco mapping acting bepveen topological spaces X and Y. Then cp is a minimal usco if and only if for each pair of open subsets U of X and W and Y with cp(U) n W 4= 0 there exists a non-empty open subset V c U such that cp(V) c W.

Theorem 3 [32]. Every fragmentable space belongs to the class of Stegall spaces. In particular, & c= class(^).

Proof. Let X be a fragmentable topological space and let d be a fragmenting metric on X. Let B be a Baire space and cp : B - • 2X be a minimal usco. For each n e N , consider the set

On:= (J {open sets V:d — diam[(p(V)] < l/n}.

Clearly each On is open. We claim that each On is dense in B. To this end, let U be a non-empty open subset of B. Since X is fragmented there exists an open set W in X, such that W n cp(U) -# 0 and d-diam[JV n cp(U)~\ < l/n. From the minimality of cp there exists a non-empty open set V of U such that cp(V) j= W. Hence d-diam[cp(V)] < l/n and so 0 4= V _= On n f/. This proves that On is dense in B.

It now follows that cp is single-valued at each point of P )n e N On; which completes

the proof. • In summary, the relationship between all these spaces is:

<H? c class(^) ^ weak Asplund spaces .= Gateaux differentiability spaces.

The question as ti whether any of these set-inclusions can be reversed is considered in the next section of this paper.

After Asplund showed that every Banach space that can be equivalently renormed to have a rotund dual norm is weak Asplund there was a considerable amount of interest in showing that every Banach space that admits an equivalent Gateaux smooth norm is weak Asplund. The fact that this would provide a true generalization of Asplund's result follows from the fact that there are Banach spaces (e.g., C[0, cOi]) with smooth norms that cannot be equivalently renormed

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to have a rotund dual norm (see [38]). (Of course it is well-known that if the dual norm is rotund then the original norm is smooth (see [10, p. 107])). The first partial solution to this was obtained by Borwein and Preiss in 1987 (see [3]) when they showed that every smooth Banach space is a Gateaux differentiability space.

Finally in 1990, 22 years after Asplund's result, it was eventually shown by Preiss, Phelps and Namioka (see [31]) that every smooth Banach space is weak Asplund.

Subsequently, this result has been improved (in 1992) of the following: "Every Banach space that admits a smooth Lipschitz bump function belongs to :#"" (see [9]). A similar result may also be found in [21].

2. Distinguishing the classes of spaces

In this section of the paper we will show that under some additional set theoretic assumptions the classes: §*, class(^) and weak Asplund spaces are distinct. The key to achieving this goal is the consideration of the following family of compact spaces.

2.1 Kalenda compacts

Let A be an arbitrary subset of (0, 1) and let

K , : = [ ( 0 , l ] x { 0 } ] u [ ( { 0 } u A ) x { l } ] .

If we equip this set with the order topology generated by the lexicographical (dictionary) ordering (i.e., (s1? s2) < (th t2) if, and only if, either s{ < tx or sx = tx and s2 < t2) then with this topology KA is a compact Hausdorff space [13, Proposition 2]. In the special case of A = (0, 1], KA reduces to the well-known

"double arrow" space.

Many of the basic properties of the Kalenda compacts may be found in [13]. In particular, the following result may be found there.

Theorem 4 [13, Proposition 3]. Let A be an arbitrary subset cf(0, 1). Then the following properties are equivalent:

(i) A is countable;

(ii) KA is metrizable;

(in) KA is fragmentable.

Hence if the set A is uncountable, then C(KA) $ & On the other hand, it is known that there are in fact uncountable sets A for which C(KA) is weak Asplund and so there are indeed examples of weak Asplund spaces that fail to belong to J?

To demonstate this we need a precise description of the duals of the C(KA) spaces.

In [20], the authors give a brief description of these dual spaces, but here, for the sake of posterity, we give a more detailed description of these dual spaces.

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2.2 A representation

The first step towards characterizing the dual space of C(KA) is to identify this space with something that looks more like C[0, 1]. (Note: Riesz's rep- resentation theorem gives a description of the dual of C(KA) in terms of regular Borel measures on KA. However, for our purposes this representation is not so useful).

Given a subset A of (0,1) we shall denote by DA the space of all real-valued functions on (0, 1] that have (i) finite right-hand limits at the points of [0, 1), (ii) are left continuous at the points of (0, 1] and (iii) are continuous at the points of (0, 1)\ A We shall consider this space endowed with the supremum norm.

Theorem 5. For every set A _Z (0, 1), DA is isometrically isomorphic to C(KA).

Proof. We define an isometry T from DA onto C(KA) as follows: T(f) ((t, 0)): = f(t) for all t e (0,1] and T(f) ((t, 1)): = lim, _>, + f(t') for t e {0} u A. One can check, as in [8, p. 47] that T is in fact an isometry from DA onto C(KA). Indeed, it is routine to verify that T is a linear isometry into C(KA), so it suffices to check that T is onto. To this end, let g e C(KA) and define / : (0, 1] -> R by f(t) : = g((t, 0))

for all t e (0, 1]. Then / e DA and T(f) = g. •

We shall now characterize the duals of these spaces in terms of functions of bounded variation. Given bounded functions / and a defined on (0, 1] and [0, 1]

respectively and a partition P : = {tk: 0 < k < n) of [0, 1] where 0 = t0 < h < t2 < ... < tn = 1,

the Riemann-Stieltjes sum of f with respect to a, determined by P, is the real number

S{P,f,a):= £/(**)•[<*(.**)-«(.*-1)]-

k = l

We say that / is Riemann-Stieltjes integrable with respect to a if there exists a real number / such that for every s > 0 there exists a partition P£ of [0, 1] such that

|5(P, / , a) — 7| < s for all partitions P that refine Pe. In this case I is denoted by,

f f{t)H'

J[0M]

.0

and is called the Riemann-Stieltjes integral of f with respect to <x.

Remark 1. The definition of the Riemann-Stieltjes integral given here differs from the standard definition in the sense that the assigned value of the function f over the subinterval \tk_u t^\ is always taken to be the evaluation at the right hand end point. The reason for this is that with the standard definition of the Riemann-Stieltjes integral, not all the functions in D(o,i) #re Riemann-Stieltjes integrable.

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For any subset A of (0,1) we shall denote by BVA[0,1] the space of all real-valued functions of bounded variation on [0,1] that are right continuous at the points of (0,1)\_4 and map 0 to 0. We will consider this space endowed with the total variation norm, i.e., for each a e -B1^[0, 1],

||a|| : = Var(a) = sup \ £ |a(t*) - a(t/c_1)|: {tk: 0 < k < n) is a partition of [0, 1] >.

Lemma 3 (Uniform approximation lemma). Let A be any dense subset of(0, 1), / e DA and 8 > 0. Then there exists a partition Pe:= [tk: 0 < k < n) of [0, 1]

with tk e A for all 1 < k < n such that \\f — / P J U < z> where fPe: (0, 1] -> IR is defined by

n

fpffi-= IfW-^-MdW-

k=\

One can now use this lemma to prove the following theorem.

Theorem 6 [20, Theorem 1]. Suppose that a : [0, 1] -> R has bounded variation and f e D(0, i). Then f is Riemann-Stieltjes integrable with respect to cc.

Proof. In order to show that / is Riemann-Stieltjes integrable with respect to a we need only show that for every 8 > 0 there exists a partition Pe of [0,1] such that

|S(Pe, / , a) - S(F, f, a)| < 8 for all partitions P' that refine P£.

An elementary calculation shows that for any g, g' e D(0 ^ and partition P we have that \S(P, g, a) - S(P, g', a)| < ||g - g'\\ • Var(a). Therefore, if we fix e > 0 and choose a partition P of [0, 1] such that \\f — fP\\ < ^/(Var(a) + 1), then

\S(P, f a) - S(F, f a)| < \S(P, f a) - S(P,fP, a)|

+ \S(P,fP,a)-S(F,f,a)\

+ \S(F,fP,a)-S(F,f,a)\

< 0 + 0 + 8 = £

for all partitions P' that refine P. • Theorem 7 [20, Theorem 2]. Let A be any subset of (0, 1). Then the dual of

DA is isometrically isomorphic to BVA[0, 1].

Proof. Consider the mapping TA : BVA[0,1] -> D* defined by, TA(a) (x):= I x(t) da(t) for each x e DA.

J [0,1]

We claim that TA is an isometry from £J^[0, 1] onto D% First let us note that TA

really does map into D% Indeed, for any given a e BVA\0, 1] it is easy to see that the mapping x i—> TA(a) (x) is linear and it is also reasonable routine to check that 17^(a) (x)| < ||a|| • ||x|| for all x G DA. Therefore, for each a e BVA[0, 1], TA(a) e D%

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In fact ||Xi(a)|| < ||a||. It should also be clear that TA is a bounded linear operator.

We will now show that TA is onto. To this end, let x* e D%

By the Hahn Banach extension theorem there exists a y*eD(*i) such that 11**11 = lly*ll a11d y*|Dx = ** (s1n c e DA is a subspace of D(0,i)). Now from Theorem 3 in [25] there exists an a e BV(0tl)[0, 1] such that j * = J-0 ^ x(t) doc(t) for all XGD(0,i) and ||a|| = \\y*\\. Unfortunately, this a may not lie in £J^[0, 1]. So we must consider a new function a : [0, 1] -» U defined by,

&(t): = a(t) if t e {0}u A u {1} and 6t(t): = lim a(t') if t e (0,1)\,4.

Then one can check, as in Theorem 13.2 of [2] that ||a|| < ||a|| and f x(t) dot(t) = \ x(t) da(t) = y*(x) = x*(x) for all x e DA.

J [0,1] J [0,1]

Therefore, TA(&) = x* and ||x*|| = \\TA(&)\\ < ||a|| = ||y*|| = ||x*||. Hence TA is

an isometry onto D*. • For a non-empty subset A of (0, 1) we shall denote by TA the topology (on

-BJ^[0,1]) of pointwise convergence on A u {1}.If A is dense in (0,1), then TA is a Hausdorff topology. Moreover, the closed unit ball in £J^[0, 1] (with respect to the total variation norm) is r^-compact.

Theorem 8 [20, Corollary 1]. For a non-empty subset A of(0, 1), (#J^[0, 1], TA) is homeomorphic to D* endowed with the weak topology generated by the functions X(o,a) with a e A u {l}. If A is dense in (0, 1), then TA is Hausdorff and the closed unit ball BBVA^^ in BVA[0,1] with the TA-topology is homeomorphic to (BDA, weak*).

In fact the mapping T defined in the previous theorem restricted to the ball BBVA[0ly realizes such a homeomorphism.

Proof. The proof of the first assertion is based upon the fact that for each oceBVA[0, 1] and teAu {1}, T(a)(x{0,q) = oc(t). The fact that T restricted to

£BV4[o,i> realizes a homeomorphism onto (BDA, weak*), follows from the fact that on BDA the relative weak* topology and the relative topology generated by the

functions X{o,q> t e A u {1} coincide (see Lemma 3). •

2.3 Distinguishing the classes of C(KJ spaces

Equipped with the representation theorem for C(KA)* we may now present the following theorem which will enable us to distinguish the classes: &, class^) and weak Asplund spaces.

Theorem 9 [14, Proposition], Let ^ be a class of Baire metric spaces which is closed with respect to taking open subspaces and dense Baire subsets and suppose that A is dense in (0, 1). Then the following assertions are equivalent:

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(i) (C(K^)*, weak*) is in Stegall's class with respect to <&;

(ii) KA is in Ste gall's class with respect to (€;

(iii) For any Be%> and any continuous function f : B -> A the function f has at least one local minimum or local maximum;

(iv) For any B e ^ and any continuous function f : B -> A there is a non-empty open set U c B such that f is constant on U.

Proof, (i) => (ii). This follows from the fact that KA is homeomorphic to a closed subspace of (C(KA)*, weak*).

(ii) => (iii). Suppose (ii) holds. Let B e%? and let / : B -> A be a continuous function. In order to obtain a contradiction, let us assume that / has no local extrema. Then the mapping cp:B —> 2KA defined by, <p(t) := {f(t)} x {0,1} is not only an usco but in fact a minimal usco. Therefore, since KA is in the class of Stegall spaces with respect to ^ we have our desired contradiction since cp is everywhere two-valued.

(iii) => (iv). Suppose that (iii) holds and that there is some Bec€ and some continuous function / : B -> A that is not constant on any non-empty open subset of B. Fix a metric p generating the topology of B. For each neN define,

£-a x : = {beB: f(b) = max {/(&'): p(b, b') < l/n}};

ET : = {beB: f(b) = min \f(b'): p(b, V) < l/n}}.

Then clearly both of the sets £™ax and Emm are closed and £ : = [jnsN £™ax u Emm

is the set of all local extrema of / on B. If one of the sets £™ax or Emm has an interior point, then / is constant on a neighbourhood of it. Indeed, if b is an interior point of £™ax then Bp(b\ 8) c £™x for some 0 < 3 < l/n. Let b' e B(b; 5). Then both f(b) > f(b') and f(b') > f(b) hold and so f(b) = /(&'); which shows that / is constant on B(b; 5). Hence both of the sets Emax and Emm are closed and nowhere dense. Therefore £ is a first category set and B' : = B\E is a dense Baire subspace of B and so it belongs to <&. Thus, by (iii), f\ff has a local extremum at a point b. Then, by continuity of / and density of B' in B, f has a local extremum at b, with respect to B, too. Thus b e E and hence we have a contradiction,

(iv) => (i). This is shown in [20, Theorem 4]. • Let ^ be a class of Baire metric spaces which is closed with respect to taking

open subsets and dense Baire subspaces. We will say that a subset A of (0, 1) satisfies property (*) with respect to <£ if for every B e^ and every continuous function / ' : B -> A there exists a non-empty open set U of B such that / is constant on U.

Corollary 1 [14, Theorem 1].

(i) If there is an uncountable dense subset A of (0, 1) that satisfies property (*) with respect to the class of all Baire metric spaces then C(KA) belongs to class(9)) but not to &;

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(ii) If there is an uncountable dense subset A of (0, 1) that satisfies property (*) with respect to the class of all Baire metric spaces of density at most card(A) then C(K^) is a weak Asplund space that does not belong to §*;

(iii) If there is a dense subset A cf(0, 1) that satisfies property (*) with respect to the class of all Baire metric saces of density at most card(A), but not property (*) with respect to the class of all Baire metric spaces then C(K^) is a weak Asplund space that does not belong to class(5?).

Proof, (i) From Theorem 4 it follows that C(KA) <£ # . On the other hand, it is shown in [20, Theorem 3] that a Banach space X belongs to class(^) if, and only if, (X*, weak*) is in the class of Stegall spaces with respect to the class of all Baire metric spaces. The result then follows from Theorem 9.

(ii) Again from Theorem 4 it follows that C(KA) $ & To show that C(KA) is weak Asplund, we need the result [8, Theorem 3.2.2] that for a Banach space X if (X*, weak*) is in the class of Stegall spaces with respect to the class of all Baire metric spaces with density at most equal to the density of X then X is weak Asplund. The result then follows from Theorem 9 and the fact that the density of C(KA) equals card(A).

(iii) As mentioned in part (ii) if (C(K^)*, weak*) is in the class of Stegall spaces with respect to the class of all Baire metric spaces with density at most equal to the density of C(K^) then C^) is weak Asplund. The fact that C(KA) <£ class(^)

follows directly from Theorem 9. • Corollory 2. Under the assumption that there exists an uncountable dense

subset A of(0, 1) that satisfies property (*) with respect to the class of all Baire metric spaces of density at most card(A), there exists a weak Asplund space that does not admit a smooth Lipschitz bump function.

Proof. It follows from Theorem 4 that C(KA) $ & and so by [9] C(K^) does not admit a smooth Lipschitz bump function. However, as in Corollary 1 part (ii)

C(KA) is weak Asplund. •

Remark 2. Till this point, we have not dwelt upon the question of whether there are in fact subsets of (0, 1) that satisfy any of the hypotheses of Corollary 1 or 2.

For a discussion on this see [13], [14], [15] and [28]. Let us mention here though that in all cases additional set-theoretic axioms are required.

The question of whether the classes of Gateaux differentiability spaces and weak Asplund spaces coincide still remains. In the next section we will discuss an attempt at distinguishing these classes.

3. The class of weakly Stegall s p a c e s

This section is divided into three subsections. In the first subsection we will define and study a class of topological spaces (i.e., the class of weakly Stegall

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spaces) that may help us distinguish Gateaux differentiability spaces from weak Asplund spaces. Then in subsection two, we shall use the class of weakly Stegall spaces to present our best known candidate for a Gateaux differentiability space that is not a weak Asplund space. Finally in subsection three, we shall examine when Banach spaces of the form C(K) belong to class(vvy?) i.e., when (C(K)*, weak*) belongs to the class of weakly Stegall spaces.

3.1 Basic properties of weakly Stegall spaces

We say that a topological space Y belongs to the class of weakly Stegall spaces if for every complete metric space M and minimal usco cp : M -i• 2y, cp is single-valued at some point of M. Although the definition of being weakly Stegall is very similar to that of being Stegall, the class of weakly Stegall spaces admit an internal characterization unlike the (known) situation for the class of Stegall spaces. To obtain this characterization we need to consider the following two player topological game.

Let X be a topological space. On X we consider the ^(X)-game played between two players 2 and Q. Player 2 goes first and chooses a non-empty subset Ax of X. Player Q must then respond by choosing a non-empty relatively open subset Bx _= A{. Following this, player 2 must select another non-empty subset A2_\Bx_\Al and in turn player Q must again select a non-empty relatively open subset B2 of A2. Continuing this procedure indefinitely the players 2 and Q produce a sequence {(4,, Bn): n e N} of pairs of non-empty subsets called a play of the ^(K)-game. We shall declare that Q wins a play {(4,, Bn):ns N} if, f)?=l Bn

is at most one point. Otherwise the player 2 is said to have won the play. By a strategy for the player 2 we mean a "rule" that specifies each move of the player 2 in every possible situation. More precisely, a strategy t := (tn: n e N) for 2 is a sequence of set-valued mappings such that 0 4= tn+l(Bh B2,..., Bn) _= Bn for all neN. The domain of each mapping tn is precisely the set of all finite sequences {Bh B2,..., Bn_x} of length n — 1 with each Bj being a non-empty relatively open subset of tj(Bh L?2,..., Bj_i) for all 1 < j < n — 1. (Note: the sequence of length 0 will be denoted by 0). Such a finite sequence {Bu B2,..., Bn_{} (infinite sequence {Bn: n e N}) is called a partial t-play (t-play). A strategy t: = (tn:ne N) for the player 2 is called a winning strategy if each t-play is won by 2 . We will say that the ^(X)-game on X is ^-unfavourable if the player 2 does not have a winning strategy in this game.

Our characterization of weakly Stagall spaces also relies upon the following topological notion.

We say that a subset Y of a topological space (X, z) has countable separation in X if there is a countable family {On: n e N} of open subsets of X such that for every pair {x,y} with y e Y and x e X\ Y, {x,y} n On is a singleton for at least one neN. For a completely regular topological space (X, T) we shall simply say

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that X has countable separation if in some compactification bX, X has countable separation in bX. It is shown in [19] that if X has countable separation in one compactification then X has countable separation in every compactification and that every Cech-analytic space has countable separation.

Theorem 10 [17, Theorems 4, 5 and 6]. For a completely regular topological space (X, T) with countable separation the following properties are equivalent:

(i) (X, T) is a weakly Stegall space;

(ii) the &(X)-game on X is H-unfavourable;

(iii) every minimal usco mapping cp : M —> 2X acting from a complete metric space M into X is single-valued at the points of a dense Baire subspace of M.

Remark 3 [26, Theorem 2]. If (X, z) is of weight K > K0, then the conditions (i) through to (iii) are also equivalent to: (iv) every minimal usco mapping cp : M -> 2X acting from a complete metric space M of density at most K into X is single-valued at some point of M.

The previous theorem enables us to establish the connection between weakly Stegall spaces and fragmentable spaces. In [18] the authors show that a space (X, T) is fragmentable if, and only if, the player CI has a winning strategy in the

^(X)-game played on X. This contrasts with the situation for weakly Stegall spaces which are characterized by the lack of a winning strategy for the player H.

Hence the distinction between being fragmentable and being weakly Stegall is equivalent to the distinction between Q having a winning strategy and S not having a winning strategy.

The condition that (X, z) has countable separation in Theorem 10 can be relaxed to "(X, T) is game determined" (see [17] for the definition of game determined spaces).

The class of weakly Stegall spaces enjoy permanence properties similar to the class of Stegall spaces. To describe and prove these properties we will need the following definition and proposition.

For a topological space X we shall denote by Qj(X) the smallest cr-algebra of subsets on X that is stable under the Souslin operation and contains all the open subsets of X.

The following result is a consequence of Proposition 5.1 and Proposition 5.3 of [36].

Proposition 1. Let cp : B -> 2X be a minimal usco acting from a Baire space B into a topological space X.IfKe S)(X) and (p~\K): = {b e B : cp(b) n K #= 0} is second category then there exist a non-empty open subset U of B and a dense Gs

subset G of U such that cp(G) <~\ K.

Theorem 11. Let (X, T) be a topological space.

(i) Let f: X -> Y be a perfect mapping onto Y. If X belongs to the class of weakly Stegall spaces then so does Y.

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(ii) Let f : X -» Y be a continuous one-to-one mapping. If Y belongs to the class of weakly Stegall spaces then so does X.

(iii) Let {Xn : n e N} be a cover of X. If each Xn e Q)(X) and belongs to the class of weakly Stegall spaces then X also belongs to the class of weakly Stegall spaces.

(iv) If X belongs to the class of weakly Stegall spaces and Y belongs to the class of Stegall spaces then the product X x Y belongs to the class of weakly Stegall spaces.

Proof. The proofs of (i), (ii) and (iv) are similar to that given in Theorem 3.1.5 of [8]. So it remains to prove (iii). Let M be a complete metric space and let cp : M -> 2X be a minimal usco. For each ne N, let Mn := cp~\Xn) := {me M : cp(m) n Xn 4= 0}. Then M = (J^°=1 Mn and so there is some nQeN such that Mno

is second category. Therefore by Proposition 1, there exist a non-empty open subset U of M and a dense G$ subset G c= U such that cp(G) ^ Xno. By Alexandroff s Theorem (see [16, p. 208]) G is completely metrizable and by Lemma 2 cp|G is a minimal usco. Therefore as Xno is a weakly Stegall space there

is some point me G such that (p(m) is a singleton. • Next we give some sufficiency conditions for a topological space to belong to

the class of weakly Stegall spaces.

Lemma 4. Let cp : X -> 2X be a minimal usco acting from a topological space X into a topological space Y. If U is a non-empty open subset of X and {Ck: 1 < k < n} is a family of closed subsets of Y such that cp(U) c= (JiUi Ck then there exist a k0 e {1,2,..., n} and a non-empty open subset V of U such that

<P{V) = ck0.

Proof. For each k e {1,2,..., n}9 let Uk:= {teU: <p(t) n Ck + 0}, then {Uk: 1 < k < n} is a closed cover of U. Hence for some k0e{l,2,..., n}, V:= int(Uko) 4- 0. It now follows from Lemma 2 that cp(V) c Cko. •

The following lemma is an immediate consequence of Stone's well-known

"lattice formulation" of the Stone-Weierstrass theorem (see [16, p. 244] or [37]).

Lemma 5. Let K be a compact subset of a topological space T and let xp denote the topology (on C(T)) ofpointwise convergence on T.IfL is a sublattice of C(T) and f e Ep then for each e > 0 there exist an lee L and an open set Ue containing K such that \f(k) — le(k)\ < sfor all k e Ue.

A topological space K is said to be countably determined if there exists an usco mapping i/> from a separable metric space P onto K, i.e., K = *A(P).

Theorem 12. Let K be a countably determined topological space. Then CP(K) belongs to the class of weakly Stegall spaces.

Proof. Let cp : M -> 2CW be a minimal usco acting from a complete metric space M into C(K). We will show that there is a point t^e M such that (p(tao) is

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singleton. Since K is a countably determined space there exists a separable metric space P (with countable base {[/„:neN}) and an usco mapping \jj : P -> 2K from P onto K, i.e., K = \//(P). For each neNwe shall define pn: C(K) -> R u {00} by,

Pn(/):=sup{|/(k)|:kGi/t(C/„)}

and Bn _= C(K) by, Bn: = {/ e C(K): pn(/) < 1}. (Note: each Bn is ^-closed and convex). We shall also let a: N -* N be any mapping onto t%l such that for each n e N o \n) is cofinal in N. We now proceed inductively.

Step 1: Choose / e cp(V0\ with V0:= M and let Lj := </> = {/} (i.e., the lattice generated by / ) . Then put sx: = sup {min{pa(l)(f — l):le Lx}: f e cp(V0)}. If Sj = 00, then choose a non-empty open subset V{ _= Vx _= V0 such that

diam(Vi) < 1 and cp(V,) n [L, + Ba(1)~] = 0.

Note: It is possible to do this since cp is a minimal usco, L{ + Ba(l) is rp-closed and cp(V0) £ Lx + Ba(l). Otherwise (i.e., sx < 00 and cp(V0) _= Lx + 8iBa(1)) one of the two cases must hold: (a) Sj = 0 or (b) 0 < s2 < 00. In case (a) cp(V0) _=

Li + s{Ba(x) = Lx and so there exists a non-empty open subset Vi _= Vi _= V0 such that diam(Vi) < 1 and <p(Vi) = {k} for some lxeLx.

In case (b) we may choose a non-empty open subset W _= Jo- such that

<p(W)n[L1 + (Sl/2)Bff(1)]=0.

Note: It is possible to do this since cp is a minimal usco, Lx + (8i/2) L?CT(1) is

^-closed and cp(V0) <£ Lx + (si/2) 5ff }. Now by Lemma 4 there exists a non-empty open subset Vi _= Vi _= PV _= 1^ such that

diam(Vi) < 1 and <p(Vi) -= /1 + Si_Jff(1) for some \xeLx. In both cases (a) and (b) inf \mm{pa(x^f — l):le Lj}: / e <p(Vi)} > 8i/2.

In general, suppose that we have completed the first n steps of the induction.

Then we will have constructed the sets: J^_=Tjj_= Vn_l _=...V2_=T^_= Vi such that diam(^) < 1// for all 1 < j < n and finite lattices L; := < / , / , . . . , / > with / G <p(Vj_l) for all 1 < j < n and extended real numbers {3: 1 < j < n} defined by, Sj: = sup {mm{pa(j)(f — l):le Lj}: f e Vj_x} such that either;

(i) cp(V^) n [Lj + Ba(j)] = 0 (in the case s; = 00) or

(ii) inf {mm {pa(j)(f - l):le Lj}: f ecp(V/)} > sjl and cp(V) _= /, + SjBa(j) for some /, G L; (in the case 0 < s} < 00).

Step (n + 1): Choose /+ e cp(Vn) and let Lw + 1:= < / , / , . . . , /n+1> (i.e., the latice generated by {/, / , . . . , /+ i} ; which is a finite set). Then put

sn+1: = sup {mm{pa(n+1)(f - l):le Ln+l}: f e <p(Vn)}.

If s„+j = 00 then choose a non-empty open set Vn+l _= Vn+1 _= 1^ such that diam(Vn+i) < l/(n + 1) and cp(Vn+l) n [Ln + 1 + _?ff(ii+1J = 0.

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Note: It is possible to do this since (p is a minimal usco, (p(Vn) £ Ln+1 + Ba{n+l)

and Ln + 1 + Ba{n+1) is rp-closed. Otherwise, (i.e., 8n+1 < oo and (p(Vn) .= Ln+1 + sn+1Ba(n+l)) one of the two cases must hold: (a) sn+1 = 0 or (b) 0 < sn+1 < oo. In case (a) (p(Vn) ^ Ln + 1 + sn+1Ba{n+l) = Ln + 1 and so by Lemma 4 there exists a non-empty open subset Vn+1 = Vn+1 ^ Vn such that

diam(Vn+1) < l/(w + 1) and (p(Vn+1) = {l+1} for some ln+1eLn+1. In case (b) we may choose a non-empty open subset W 9= Vn such that

<?(W)n[L„+ 1 + (Sn+1/2)Bff(+1)] = 0.

Note: This is possible since (p is a minimal usco, Ln + 1 + (sn+1/2) £<T(+1) is Tp-closed and (p(Vn) £ Ln+1 + (sn+1/2) -Bg(w+1)- Now by Lemma 4 there exists a non-empty open subset Vn+1 = Vn+1_=KV.= Vo such that

diam(Vn+1) < l/(w + 1) and (p(Vn+1) ^ /n+1 + sn+1Ba{n+l) for some ln+1eLn+1. In both cases (a) and (b) inf {min{pa{n+l)(f - l):le Ln+1}: f e (p(Vn+1)} > sn+1/2.

This completes the induction.

Let {too}:= f]n=1Vn. We claim that (p is single-valued at t^. To justify this assertion let us consider the following argument. Let fge (p(t^) and ke K. We need to show that \f(k) — g(k)\ = 0. To this end, let s be an arbitrary positive real number less that 1. Since i/t is onto there exists a p e P such that k e \j/(p). Let /oo e (pfooo) be a Tp-cluster-point of the sequence (fn:ne N). The existence of such a cluster-point follows from the fact that q> is an usco mapping. Then /oo e [fn:ne N\Tp ^ L^Xp, where L^ : = \JneN Ln. Now since i/>(p) is compact and Loo is a lattice we have by Lemma 5 that there exists an l^e L^ and an open set U containing \j/(p) such that \fo0(k!) — /00(^/)l < £/4 f°r &-- K e U.

Since \j/ is an usco there exists an w0 e N such that pno(/oo — /oo) < e/4. Next, if we choose i sufficiently large so that /^ e Lt and cr(i) = w0, then at the zth stage of the induction st < oo since f^ e (p(V) n [L, + Bff(0] -t- 0 and so

£/4 > P„0(/oo - /oo) > inf {min{pno(/ - / ) : / e Lt}: f e (p(V)} > sJ2.

Therefore 0 < st < s/2. Moreover, by the construction we have that fge (D(too) c cp(V^ c /. + SiBno c /. + (£/2) Bno

and so |/(k) — g(k)\ < e; which completes the proof. • A subset K of a topological space T is said to be relatively pseudo-compact in

T if every continuous real-valued mapping defined on T is bounded on K. It follows from Lemma 3.3 in [39] that the conclusion of Lemma 5 remains valid under the assumption that K is relatively pseudo-compact in T and L is countable.

In this way, we can see that the conclusion of Theorem 12 (i.e., CP(K) is weakly Stegall) remains valid under the weaker assumption that K is "pseudo-countably 82

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determined", i.e., there exists a second countable space P and a set-valued mapping

\j/ from P onto K such that (i) for each p e P, \j/(p) is relatively pseudo-compact in K and (ii) for each peP and open set W in K with {(/(p) ^ W there exists a neighbourhood U of p such that IA(^) — W.

Let us further note that by appealing to the game characterization of fragmen- tability given in [19] one can modify the argument in Theorem 12 to show the following: "Suppose K is pseudo-countably determined and B c C(K). If player Q has a strategy s in the ^(S)-game played on B such that for every 8-play (An: n e N) either (i) f]neN An = 0 or (ii) every sequence (fn:ne N) with fn e An

for all neN has a rp-cluster-point in CP(K), then B is fragmented by a metric d such that the d-topology is at least as strong as the relative pointwise topology on £".

3.2 Search for a GDS that is not weak A s p l u n d

It has been a long standing open question as to whether every Gateaux differentiability space is a weak Asplund space. In an attempt to make some progress on this question, the authors in [26] considered the following class of Banach spaces. We say that a Banach space X belongs to class(w&) if (K*, weak*) belongs to the class of weakly Stegall spaces. The relationship between class(w^) spaces and both weak Asplund and Gateaux differentiability spaces is revealed in the next theorem.

Theorem 13. Every member of class(w£P) has the property that every con- tinuous convex function defined on a non-empty open convex subset of it is Gateaux differentiate at each point of a second category set.

Given Theorem 13, it is natural to ask whether there is an example of a non-weak Asplund space that belongs to class(w^). Unfortunately, the construc- tion of such a Banach space remains elusive. However, below we shall present an example If a Banach space X that belongs to class(w^) but not to class(y?).

(Disappointingly, this space is known to be weak Asplund, [14]).

Example 1. If there exists an uncountable dense subset A of(0, 1) that satisfies property (*) with respect to the class of all Baire metric spaces of density at most

card(A), but not property (*) with respect to the class of all Baire metric spaces then there is a Banach space X that belongs to class(w&) but not to class(£f).

Proof. It follows from Corollary 1 that C(KA) $ class(^). However, from

Remark 3 it follows that C(KA) e class(w^). •

We note here that if a Banach space Y belongs to class^) and A _= (0, 1) satisfies property (*) with respect to the class of Baire metric spaces of density at most card(^4) then C(KA) x Y belongs to class(wy') and so is a Gateaux dif- ferentiability space. However, it is not at all clear whether C(KA) x Y is weak

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Asplund. Certainly, the argument used in [14] to show that C(KA) is weak Asplund will fail if the density of Y is large.enough (i.e., bigger than card(A)), So our best candidate for a non-weak Asplund Gateaux differentiability space is: C(KA) x Y for some Banach space Y e class(^).

3 . 3 C(K) s p a c e s t h a t b e l o n g to class(w,9b)

In the study of weak Asplund spaces, Banach spaces of the form C(KA) have received special attention. Indeed, for spaces of this form several necessary conditions (on K) are known for C(K) to be weak Asplund, e.g., every closed subspace of K must contain a dense completely metrizable subspace (see [6]).

Several sufficiency conditions are also known. For example it is known that C(K) belongs to # if, and only if, K is fragmentable (see [32]). Unfortunately the corresponding result for the class of Stegall spaces remains unknown. However, for the class of weakly Stegall spaces we have the following result.

Let B be a Bernstein subset of (0, 1), that is, B is a subset of (0, 1) such that neither B nor its complement contains any perfect compact sets (see [29, p. 23]) and put

A:= [(0,1/2) n B ] u [1/2 + (0,1/2) \ B ]

(then A does not contain any perfect compact subsets either) and then define KA as in section two.

Example 2 [26, Example 3]. With A as above, KA is a weakly Stegall space but C(KA) $ class(w£?). In fact there exists a continuous convex function defined on C(KA) whose points of Gateaux differentiability are at most first category.

Therefore an interesting question is to characterize those compact spaces K for which C(K) belongs to class(w^). In fact even those more modest question of characterizing those subsets A of (0, 1) for which C(KA) belongs to class(wty?) is both interesting and open.

R e f e r e n c e s

[1] ASPLUND E., Fréchet differentiability of convex functions, Acta Math. 121 (1968), 31—47.

[2] BACHMAN G. and NARICI L., Functional Analysis, Academic Press, New Үork, 1966.

[3] BORWEIN J. M. and PREISS D., A smooth variational principle with applications to subdifferentia- bility and to differentiability of convexfunctions, Trans. Amer. Matn. Soc. 303 (1987), 515—527.

[4] BOURGIN R. D., Geometric aspects ofconvex sets with the Radon-Nikodým property, Lecture Notes in Mathematics, 993 Springer-Verlag, Berlin, 1983.

[5] CHRІSTENSEN J. P. R., Theorems ofNamioka and R. E. Johnson typefor upper semicontinuous and compact valued set-valued mappings, Proc. Amer. Matn. Soc. 86 (1982), 649 — 655.

[6] ČOBAN M. M. and KENDEROV P. S., Generic Gâteaux differentiability of convex functionals in C(T) and the topological properties of T, Proceedings of 15th Spring Conference of the Union of Bulgarian Mathematicians, Sljančev Brjag (1986), 141 — 149.

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