凸包的Szlenk指数
Szlenk indices of convex hulls
摘要：
本文研究了定义在对偶巴拿赫空间子集上的非紧性的一般测度及其导子和$ω$ -迭代。我们引入了非紧性的可凸性和次线性测度的概念，并研究了与之相关的分片导子和片导子的性质。我们将所得结果应用于非紧性的Kuratowski测度和巴拿赫空间的Szlenk指数的研究。由此得到可分巴拿赫空间的Szlenk指标与凸Szlenk指标总是相等的.对于任意可数序数$α$，我们还利用等价重赋范的存在性给出了Szlenk指数以$ω^{α+1}$为界的巴拿赫空间的一个刻划.这推广了Knaust，Odell和Schlumprecht关于Szlenk指数等于$ω$的巴拿赫空间的一个结果.
英文摘要：
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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