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A ξ-weak Grothendieck compactness principle

Part of: Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  10 March 2021

KEVIN BEANLAND  and
RYAN M. CAUSEY
KEVIN BEANLAND
Affiliation:
Washington and Lee University, 204 W. Washington Street, Chavis Hall Room 103, Lexington, VA 24450, U.S.A. e-mail: beanlandk@wlu.edu
RYAN M. CAUSEY
Affiliation:
123 Bachelor Hall, 301 S Patterson Avenue, Oxford, OH 45056, U.S.A. e-mail: rmcausey1701@gmail.com
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Abstract

For 0 ≤ ξ ≤ ω1, we define the notion of ξ-weakly precompact and ξ-weakly compact sets in Banach spaces and prove that a set is ξ-weakly precompact if and only if its weak closure is ξ-weakly compact. We prove a quantified version of Grothendieck’s compactness principle and the characterisation of Schur spaces obtained in [7] and [9]. For 0 ≤ ξ ≤ ω1, we prove that a Banach space X has the ξ-Schur property if and only if every ξ-weakly compact set is contained in the closed, convex hull of a weakly null (equivalently, norm null) sequence. The ξ = 0 and ξ= ω1 cases of this theorem are the theorems of Grothendieck and [7], [9], respectively.

MSC classification

Primary: 46B03: Isomorphic theory (including renorming) of Banach spaces
Secondary: 46B20: Geometry and structure of normed linear spaces

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 172 , Issue 1 , January 2022 , pp. 231 - 246
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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