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1 November 2017 A metric interpretation of reflexivity for Banach spaces
P. Motakis, T. Schlumprecht
Duke Math. J. 166(16): 3001-3084 (1 November 2017). DOI: 10.1215/00127094-2017-0021

Abstract

We define two metrics d1,α and d,α on each Schreier family Sα, α<ω1, with which we prove the following metric characterization of the reflexivity of a Banach space X: X is reflexive if and only if there is an α<ω1 such that there is no mapping Φ:SαX for which cd,α(A,B)Φ(A)Φ(B)Cd1,α(A,B)for allA,BSα. Additionally we prove, for separable and reflexive Banach spaces X and certain countable ordinals α, that max (Sz(X),Sz(X))α if and only if (Sα,d1,α) does not bi-Lipschitzly embed into X. Here Sz(Y) denotes the Szlenk index of a Banach space Y.

Citation

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P. Motakis. T. Schlumprecht. "A metric interpretation of reflexivity for Banach spaces." Duke Math. J. 166 (16) 3001 - 3084, 1 November 2017. https://doi.org/10.1215/00127094-2017-0021

Information

Received: 2 May 2016; Revised: 3 April 2017; Published: 1 November 2017
First available in Project Euclid: 21 September 2017

zbMATH: 06812214
MathSciNet: MR3715804
Digital Object Identifier: 10.1215/00127094-2017-0021

Subjects:
Primary: 46B03
Secondary: 46B10 , 46B80

Keywords: metric characterization of Banach spaces , reflexivity , Schreier families , Szlenk index

Rights: Copyright © 2017 Duke University Press

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Vol.166 • No. 16 • 1 November 2017
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