Szlenk indices of convex hulls

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• 作者：G. Lancien^a ; ^{gilles.lancien@univ-fcomte.fr" class="auth_mail" title="E-mail the corresponding author} ; A. Prochá ; zka^a ; ^{antonin.prochazka@univ-fcomte.fr" class="auth_mail" title="E-mail the corresponding author} ; M. Raja^b ; ^{matias@um.es" class="auth_mail" title="E-mail the corresponding author}
• 关键词：46B20
• 刊名：Journal of Functional Analysis
• 出版年：2017
• 出版时间：15 January 2017
• 年：2017
• 卷：272
• 期：2
• 页码：498-521
• 全文大小：494 K

文摘

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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616303172&_mathId=si1.gif&_user=111111111&_pii=S0022123616303172&_rdoc=1&_issn=00221236&md5=2e327c660db228b0482b2ca54312a343" title="Click to view the MathML source">ω^α+1class="mathContainer hidden">class="mathCode">${\omega }^{\alpha +1}$ 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.