A Gâteaux differentiability space that is not weak Asplund

Moors, Warren B.; Somasundaram, S.

doi:10.1090/s0002-9939-06-08402-4

Cited by 17 publications

(4 citation statements)

References 6 publications

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“…A Banach space X is called a Gâteaux differentiability space if every convex continuous function defined on a non-empty open convex subset D of X is Gâteaux differentiable at densely many points of D. This notion differs from the notion of weak Asplund space only by virtue of the fact that the set of differentiability points is not required to contain a dense G δ , but merely to be dense in D (for information concerning those spaces, consult [12,61]). On the other hand, let us stress the fact that the two notions are actually distinct, as it was first proved in [57]. Let us also refer to [55] for a simplified proof of the example and to [36,39,41,56] for related results.…”

Section: 1mentioning

confidence: 94%

Separated sets and Auerbach systems in Banach spaces

Hájek

Kania

Russo

2020

Trans. Amer. Math. Soc.

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The paper elucidates the relationship between the density of a Banach space and possible sizes of Auerbach systems and well-separated subsets of its unit sphere. For example, it is proved that for a large enough space X, the unit sphere S X always contains an uncountable (1+)-separated subset. In order to achieve this, new results concerning the existence of large Auerbach systems are established, that happen to be sharp for the class of WLD spaces. In fact, we offer the first consistent example of a non-separable WLD Banach space that contains no uncountable Auerbach system, as witnessed by a renorming of c 0 (ω 1 ). Moreover, the following optimal results for the classes of, respectively, reflexive and super-reflexive spaces are established: the unit sphere of an infinite-dimensional reflexive space contains a symmetrically (1 + ε)-separated subset of any regular cardinality not exceeding the density of X; should the space X be superreflexive, the unit sphere of X contains such a subset of cardinality equal to the density of X. The said problem is studied for other classes of spaces too, including WLD spaces, RNP spaces, or strictly convex ones.

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Section: 1mentioning

confidence: 94%

Separated sets and Auerbach systems in Banach spaces

Hájek

Kania

Russo

2020

Trans. Amer. Math. Soc.

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“…Known examples [38][39][40] of individual convex continuous functions that are Gateaux-differentiable on a dense, but non-residual, subset of their domain suggest that a Gateaux-differentiability space is not necessarily a weak Asplund space. A first answer to Larman and Phelps's question has been given in 2006, in [41] where a Gateaux-differentiability space X that is not a weak Asplund space is constructed. This shows that Theorem 4.5 and Corollary 4.6 are very general, probably optimum, results on the weak * -density of the set of exposed points in the extreme boundary of a convex weak * -compact set K ∈ CK(X * ).…”

Section: Corollary 46 (Extension Of the Straszewicz Theorem -Ii)mentioning

confidence: 99%

Weak$^{\ast }$ Hypertopologies with Application to Genericity of Convex Sets

Bru¹,

Pedra²

2021

Preprint

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We propose a new class of hypertopologies, called here weak * hypertopologies, on the dual space X * of a real or complex topological vector space X . The most well-studied and wellknown hypertopology is the one associated with the Hausdorff metric for closed sets in a complete metric space. Therefore, we study in detail its corresponding weak * hypertopology, constructed from the Hausdorff distance on the field (i.e. R or C) of the vector space X and named here the weak * -Hausdorff hypertopology. It has not been considered so far and we show that it can have very interesting mathematical connections with other mathematical fields, in particular with mathematical logics. We explicitly demonstrate that weak * hypertopologies are very useful and natural structures by using again the weak * -Hausdorff hypertopology in order to study generic convex weak * -compact sets in great generality. We show that convex weak * -compact sets have generically weak * -dense set of extreme points in infinite dimensions. An extension of the wellknown Straszewicz theorem to Gateaux-differentiability (non necessarily Banach) spaces is also proven in the scope of this application.

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“…In 2006, Waren B. Moors and Sivajah Somasundaram proved that there exists a Gâteaux differentiable space that is not a weak Asplund space (see [2]). In 1979, D.G.…”

Section: Introductionmentioning

confidence: 99%

Gateaux Differentiability of Convex Functions and Weak Dentable Set in Nonseparable Banach Spaces

Shang

Cui

2019

Journal of Function Spaces

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In this paper, we prove that if C⁎⁎ is a ε-separable bounded subset of X⁎⁎, then every convex function g≤σC is Ga^teaux differentiable at a dense Gδ subset G of X⁎ if and only if every subset of ∂σC(0)∩X is weakly dentable. Moreover, we also prove that if C is a closed convex set, then dσC(x⁎)=x if and only if x is a weakly exposed point of C exposed by x⁎. Finally, we prove that X is an Asplund space if and only if, for every bounded closed convex set C⁎ of X⁎, there exists a dense subset G of X⁎⁎ such that σC⁎ is Ga^teaux differentiable on G and dσC⁎(G)⊂C⁎. We also prove that X is an Asplund space if and only if, for every w⁎-lower semicontinuous convex function f, there exists a dense subset G of X⁎⁎ such that f is Ga^teaux differentiable on G and df(G)⊂X⁎.

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A Gâteaux differentiability space that is not weak Asplund

Abstract: Abstract. In this paper we construct a Gâteaux differentiability space that is not a weak Asplund space. Thus we answer a question raised by David Larman and Robert Phelps from 1979.

Cited by 17 publications

References 6 publications

Separated sets and Auerbach systems in Banach spaces

Separated sets and Auerbach systems in Banach spaces

Weak$^{\ast }$ Hypertopologies with Application to Genericity of Convex Sets

Gateaux Differentiability of Convex Functions and Weak Dentable Set in Nonseparable Banach Spaces

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