{ "id": "1604.07271", "version": "v1", "published": "2016-04-25T14:21:03.000Z", "updated": "2016-04-25T14:21:03.000Z", "title": "A metric interpretation of reflexivity for Banach spaces", "authors": [ "Pavlos Motakis", "Thomas Schlumprecht" ], "categories": [ "math.FA" ], "abstract": "We define two metrics $d_{1,\\alpha}$ and $d_{\\infty,\\alpha}$ on each Schreier family $\\mathcal{S}_\\alpha$, $\\alpha<\\omega_1$, 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 $\\alpha<\\omega_1$, so that there is no mapping $\\Phi:\\mathcal{S}_\\alpha\\to X$ for which $$ cd_{\\infty,\\alpha}(A,B)\\le \\|\\Phi(A)-\\Phi(B)\\|\\le C d_{1,\\alpha}(A,B) \\text{ for all $A,B\\in\\mathcal{S}_\\alpha$.}$$ Secondly, we prove for separable and reflexive Banach spaces $X$, and certain countable ordinals $\\alpha$ that $\\max(\\text{ Sz}(X),\\text{ Sz}(X^*))\\le \\alpha$ if and only if $({\\mathcal S}_\\alpha, d_{1,\\alpha})$ does not bi-Lipschitzly embed into $X$. Here $\\text{Sz}(Y)$ denotes the Szlenk index of a Banach space $Y$.", "revisions": [ { "version": "v1", "updated": "2016-04-25T14:21:03.000Z" } ], "analyses": { "subjects": [ "46B03", "46B10", "46B80" ], "keywords": [ "metric interpretation", "reflexivity", "reflexive banach spaces", "metric characterization" ], "note": { "typesetting": "TeX", "pages": 0, "language": "en", "license": "arXiv", "status": "editable", "adsabs": "2016arXiv160407271M" } } }