Some remarks on Radon–Nikodym compact spaces

Abstract. σ-Asplund generated Banach spaces are used to give new characterizations of subspaces of weakly compactly generated spaces and to prove some results on Radon-Nikodým compacta. We show, typically, that in the framework of weakly Lindelöf determined Banach spaces, subspaces of weakly compactly generated spaces are the same as σ-Asplund generated spaces. For this purpose, we study relationships between quantitative versions of Asplund property, dentability, differentiability, and of weak compactness in Banach spaces. As a consequence, we provide a functional-analytic proof of a result of Arvanitakis: A compact space is Eberlein if (and only if) it is simultaneously Corson and quasi-Radon-Nikodým.

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“…The implication (iii)⇒(i) in Theorem 3 is due to Arvanitakis [2]. In his proof, he used purely topological tools.…”

Section: Theorem 1 Let X Be a Banach Space Then (O)⇒(i)⇒(ii)⇒(iv) Amentioning

confidence: 99%

Weak compactness and σ-Asplund generated Banach spaces

2007

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“…Arvanitakis [2] has taken the following approach to this problem: if K is a Radon-Nikodým compact and π : K → L is a continuous surjection, then we have a lower semicontinuous fragmenting metric d on K, and if we want to prove that L is Radon-Nikodým compact, we should find such a metric on L. A natural candidate is…”

Section: Theorem 1 a Compact Space K Is Radon-nikodým Compact If Andmentioning

confidence: 99%

“…Arvanitakis [2] has taken the following approach to this problem: if K is a Radon-Nikodým compact and π : K → L is a continuous surjection, then we have a lower semicontinuous fragmenting 2000 Mathematics Subject Classification: Primary 46B26; Secondary 46B22, 46B50, 54G99.…”

Section: Compact Space K Is Radon-nikodým Compact If and Only If Thermentioning

confidence: 99%

“…Techniques of Arvanitakis [2] play an important role in this section, as well as the following theorem of Namioka [10]: Theorem 9. Let K be a compact space.…”

Section: Quasi-radon-nikodým Compacta Of Low Weightmentioning

confidence: 99%

“…But it is not a metric because, in general, it fails the triangle inequality. Consequently, Arvanitakis [2] introduced the following concept:…”

Section: Theorem 1 a Compact Space K Is Radon-nikodým Compact If Andmentioning

confidence: 99%

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Radon–Nikodým compact spaces of low weight and Banach spaces

Avilés

2005

Studia Math.

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Abstract. We prove that a continuous image of a Radon-Nikodým compact of weight less than b is Radon-Nikodým compact. As a Banach space counterpart, subspaces of Asplund generated Banach spaces of density character less than b are Asplund generated. In this case, in addition, there exists a subspace of an Asplund generated space which is not Asplund generated and which has density character exactly b.The concept of Radon-Nikodým compact, due to Reynov [13], has its origin in Banach space theory, and it is defined as a topological space which is homeomorphic to a weak * compact subset of the dual of an Asplund space, that is, a dual Banach space with the Radon-Nikodým property (topological spaces will be here assumed to be Hausdorff). In [10], the following characterization of this class is given: compact space K is Radon-Nikodým compact if and only if there is a lower semicontinuous metric d on K which fragments K.Recall that a map f : X × X → R on a topological space X is said to fragment X if for each (closed) subset L of X and each ε > 0 there is a nonempty relatively open subset U of L of f -diameter less than ε, i.e. sup{f (x, y) : x, y ∈ U } < ε. Also, a map g : Y → R from a topological space to the real line is lower semicontinuous if {y : g(y) ≤ r} is closed in Y for every real number r.It is an open problem whether a continuous image of a Radon-Nikodým compact is Radon-Nikodým. Arvanitakis [2] has taken the following approach to this problem: if K is a Radon-Nikodým compact and π : K → L is a continuous surjection, then we have a lower semicontinuous fragmenting 2000 Mathematics Subject Classification: Primary 46B26; Secondary 46B22, 46B50, 54G99.

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