{"paper":{"title":"Bernstein's Lethargy Theorem in Frechet Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Asuman Guven Aksoy, Grzegorz Lewicki","submitted_at":"2015-03-20T18:43:03Z","abstract_excerpt":"In this paper we consider Bernstein's Lethargy Theorem (BLT) in the context of Fr\\'{e}chet spaces. Let $X$ be an infinite-dimensional Fr\\'echet space and let $\\mathcal{V}=\\{V_n\\}$ be a nested sequence of subspaces of $ X$ such that $ \\bar{V_n} \\subseteq V_{n+1}$ for any $ n \\in \\mathbb{N}$ and $\nX=\\bar{\\bigcup_{n=1}^{\\infty}V_n}.$\nLet $ e_n$ be a decreasing sequence of positive numbers tending to 0. Under an additional natural condition on $\\sup\\{\\{dist}(x, V_n)\\}$, we prove that there exists $ x \\in X$ and $ n_o \\in \\mathbb{N}$ such that $$ \\frac{e_n}{3} \\leq \\{dist}(x,V_n) \\leq 3 e_n $$ for "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.06190","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"}