{"paper":{"title":"A $\\xi$-weak Grothendieck compactness principle","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Kevin Beanland, R.M. Causey","submitted_at":"2019-05-29T13:44:06Z","abstract_excerpt":"For $0\\leqslant \\xi\\leqslant \\omega_1$, we define the notion of $\\xi$-weakly precompact and $\\xi$-weakly compact sets in Banach spaces and prove that a set is $\\xi$-weakly precompact if and only if its weak closure is $\\xi$-weakly compact. We prove a quantified version of Grothendieck's compactness principle and the characterization of Schur spaces obtained by Dowling et al. For $0\\leqslant \\xi\\leqslant \\omega_1$, we prove that a Banach space $X$ has the $\\xi$-Schur property if and only if every $\\xi$-weakly compact set is contained in the closed, convex hull of a weakly null (equivalently, no"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.12455","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"}