{"paper":{"title":"A metric interpretation of reflexivity for Banach spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.FA","authors_text":"Pavlos Motakis, Thomas Schlumprecht","submitted_at":"2016-04-25T14:21:03Z","abstract_excerpt":"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}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.07271","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}