{"paper":{"title":"Constructing an Element of a Banach Space with Given Deviation from its Nested Subspaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Asuman G. Aksoy, Qidi Peng","submitted_at":"2016-05-15T18:50:10Z","abstract_excerpt":"This paper contains two improvements on a theorem of S. N. Bernstein for Banach spaces. We show that if $X$ is an arbitrary infinite-dimensional Banach space, $\\{Y_n\\}$ is a sequence of strictly nested subspaces of $ X$ and if $\\{d_n\\}$ is a non-increasing sequence of non-negative numbers tending to 0, then for any $c\\in(0,1]$ we can find $x_{c} \\in X$, such that the distance $\\rho(x_{c}, Y_n)$ from $x_{c}$ to $Y_n$ satisfies $$ c d_n \\leq \\rho(x_{c},Y_n) \\leq 4c d_n,~\\mbox{for all $n\\in\\mathbb N$}. $$ We prove the above inequality by first improving Borodin (2006)'s result for Banach spaces b"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.04592","kind":"arxiv","version":4},"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"}