A metric interpretation of reflexivity for Banach spaces

P. Motakis, T. Schlumprecht
2017 Duke mathematical journal  
We define two metrics d_1,α and d_∞,α on each Schreier family S_α, α<ω_1, with which we prove the following metric characterization of reflexivity of a Banach space X: X is reflexive if and only if there is an α<ω_1, so that there is no mapping Φ:S_α→ X for which cd_∞,α(A,B)<Φ(A)-Φ(B)< C d_1,α(A,B) for all A,B∈S_α. Secondly, we prove for separable and reflexive Banach spaces X, and certain countable ordinals α that ( Sz(X), Sz(X^*))<α if and only if ( S_α, d_1,α) does not bi-Lipschitzly embed
more » ... to X. Here Sz(Y) denotes the Szlenk index of a Banach space Y.
doi:10.1215/00127094-2017-0021 fatcat:exopt7ujkrbvxhmbf6hfrv3lqy