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arXiv · 1905.12455

A $\xi$-weak Grothendieck compactness principle

Kevin Beanland·R. M. Causey
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Abstract

For $0\leqslant \xi\leqslant \omega_1$, we define the notion of $\xi$-weakly precompact and $\xi$-weakly compact sets in Banach spaces and prove that a set is $\xi$-weakly precompact if and only if its weak closure is $\xi$-weakly compact. We prove a quantified version of Grothendieck's compactness principle and the characterization of Schur spaces obtained by Dowling et al. For $0\leqslant \xi\leqslant \omega_1$, we prove that a Banach space $X$ has the $\xi$-Schur property if and only if every $\xi$-weakly compact set is contained in the closed, convex hull of a weakly null (equivalently, norm null) sequence.

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BibTeXRIS

Kevin Beanland, R. M. Causey. 2019-05-29. A $\xi$-weak Grothendieck compactness principle. https://arxiv.org/abs/1905.12455

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