Some more recent results concerning weak Asplund spaces

Moors, Warren B.

doi:10.1155/aaa.2005.307

Cited by 7 publications

(4 citation statements)

References 5 publications

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“…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: 95%

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: 95%

Separated sets and Auerbach systems in Banach spaces

Hájek

Kania

Russo

2020

Trans. Amer. Math. Soc.

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“…We will consider this space endowed with the total variation norm, i.e., for each α ∈ BV A [0, 1], A more detailed analysis of Proposition 1 may be found in [9,Theorem 7]. Next, we give some technical results that will be needed in our main theorem.…”

Section: A Weakly Stegall Space That Is Not Weak Asplundmentioning

confidence: 99%

“…Since this manuscript was first submitted in 2002, a simplified proof of Corollary 2 has appeared in [8].…”

Section: Note Added In Proofmentioning

confidence: 99%

A Gâteaux differentiability space that is not weak Asplund

Moors¹,

Somasundaram²

2006

Proc. Amer. Math. Soc.

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“…Interestingly, uncountable subsets of R that do not contain any perfect compact subsets played an important role in the construction of (i) a Gâteaux differentiability space that is not weak Asplund [13,14,27,29,30], (ii) a dual differentiation space that does not admit an equivalent locally uniformly rotund norm [22] and (iii) a Namioka space without an equivalent Kadeč norm [31].…”

mentioning

confidence: 99%

Fragmentability by the Discrete Metric

Moors¹

2015

Bull. Aust. Math. Soc.

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In a recent paper, topological spaces (X, τ) that are fragmented by a metric that generates the discrete topology were investigated. In the present paper we shall continue this investigation. In particular, we will show, among other things, that such spaces are σ-scattered, that is, a countable union of scattered spaces, and characterise the continuous images of separable metrisable spaces by their fragmentability properties.2010 Mathematics subject classification: primary 46B20; secondary 46B22.

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Some more recent results concerning weak Asplund spaces

Abstract: In this paper, we will present some of the latest advances that have occurred in the study of weak Asplund spaces. In particular, we will give an example of a GÃƒÂ¢teaux differentiability space that is not weak Asplund

Cited by 7 publications

References 5 publications

Separated sets and Auerbach systems in Banach spaces

Separated sets and Auerbach systems in Banach spaces

A Gâteaux differentiability space that is not weak Asplund

Fragmentability by the Discrete Metric

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