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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\\"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1606.07977","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-06-25T23:48:20Z","cross_cats_sorted":[],"title_canon_sha256":"e84d42e1e2ac6e4bd011a37980633562feb0fd1bbab8eed8cd15a7f00a788aa0","abstract_canon_sha256":"f0fe4d2cad91e8d4c488f888cf2cd4bad8c462477a9e529eff9baa008bb16ad3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:03:36.221890Z","signature_b64":"OWP0RyhO72K/qYP100l4WQsZIK/I9DI6iGMcnmWf2sS3+BBtFYaabMjE9Kn+s6/uG++pOE01y8MSQVjb1JsmDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9415fcb1d309410e47d90a6f7224b189fd26382b78d39b64470b2dfd117f9aab","last_reissued_at":"2026-05-18T01:03:36.221363Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:03:36.221363Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $n\\ge1$. 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