Duke Math. J. 166 (16), 3001-3084, (1 November 2017) DOI: 10.1215/00127094-2017-0021
P. Motakis, T. Schlumprecht
KEYWORDS: metric characterization of Banach spaces, reflexivity, Szlenk index, Schreier families, 46B03, 46B10, 46B80
We define two metrics and on each Schreier family , , with which we prove the following metric characterization of the reflexivity of a Banach space : is reflexive if and only if there is an such that there is no mapping for which Additionally we prove, for separable and reflexive Banach spaces and certain countable ordinals , that if and only if does not bi-Lipschitzly embed into . Here denotes the Szlenk index of a Banach space .