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Article Dans Une Revue Journal of Functional Analysis Année : 2017

Szlenk indices of convex hulls

Résumé

We study the general measures of non-compactness defined on subsets of a dual Banach space, their associated derivations and their ω-iterates. We introduce the notions of convexifiable and sublinear measure of non-compactness and investigate the properties of its associated fragment and slice derivations. We apply our results to the Kuratowski measure of non-compactness and to the study of the Szlenk index of a Banach space. As a consequence, we obtain that the Szlenk index and the convex Szlenk index of a separable Banach space are always equal. We also give, for any countable ordinal α, a characterization of the Banach spaces with Szlenk index bounded by ω α+1 in terms of the existence of an equivalent renorming. This extends a result by Knaust, Odell and Schlumprecht on Banach spaces with Szlenk index equal to ω.
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Dates et versions

hal-01763720 , version 1 (11-04-2018)

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Gilles Lancien, A Prochazka, M. Raja. Szlenk indices of convex hulls. Journal of Functional Analysis, 2017, 272 (2), pp.498 - 521. ⟨10.1016/j.jfa.2016.10.013⟩. ⟨hal-01763720⟩
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