{"paper":{"title":"Subspace Condition for Bernstein's Lethargy Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Asuman G\\\"uven Aksoy, Caleb Case, Monairah Al-Ansari, Qidi Peng","submitted_at":"2016-06-25T23:48:20Z","abstract_excerpt":"In this paper, we consider a condition on subspaces in order to improve bounds given in the Bernstein's Lethargy Theorem (BLT) for Banach spaces. Let $d_1 \\geq d_2 \\geq \\dots d_n \\geq \\dots > 0$ be an infinite sequence of numbers converging to $0$, and let $Y_1 \\subset Y_2 \\subset \\dots\\subset Y_n \\subset \\dots \\subset X$ be a sequence of closed nested subspaces in a Banach space $X$ with the property that $\\overline{Y}_{n}\\subset Y_{n+1}$ for all $n\\ge1$. We prove that for any $c \\in (0,1]$, there exists an element $x_c \\in X$ such that $$ c d_n \\leq \\rho(x_c, Y_n) \\leq \\min (4, \\tilde{a}) c\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.07977","kind":"arxiv","version":2},"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"}