Duke Mathematical Journal

A metric interpretation of reflexivity for Banach spaces

P. Motakis and T. Schlumprecht

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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.

Article information

Duke Math. J., Volume 166, Number 16 (2017), 3001-3084.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46B03: Isomorphic theory (including renorming) of Banach spaces
Secondary: 46B10: Duality and reflexivity [See also 46A25] 46B80: Nonlinear classification of Banach spaces; nonlinear quotients

metric characterization of Banach spaces reflexivity Szlenk index Schreier families


Motakis, P.; Schlumprecht, T. A metric interpretation of reflexivity for Banach spaces. Duke Math. J. 166 (2017), no. 16, 3001--3084. doi:10.1215/00127094-2017-0021. https://projecteuclid.org/euclid.dmj/1505959221

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