{"paper":{"title":"Banach spaces with no proximinal subspaces of codimension 2","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Charles John Read","submitted_at":"2013-07-30T13:03:44Z","abstract_excerpt":"The classical theorem of Bishop-Phelps asserts that, for a Banach space X, the norm-achieving functionals in X* are dense in X*. Bela Bollobas's extension of the theorem gives a quantitative description of just how dense the norm-achieving functionals have to be: if (x,f) is in X x X* with ||x||=||f||=1 and |1-f(x)|< h^2/4 then there are (x',f') in X x X* with ||x'||= ||f'||=1, ||x-x'||, ||f-f'||< h and f'(x')=1.\n  This means that there are always \"proximinal\" hyperplanes H in X (a nonempty subset E of a metric space is said to be \"proximinal\" if, for x not in E, the distance d(x,E) is always "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.7958","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"}