{"paper":{"title":"Grothendieck's theorem for Bessel sequences","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.FA","authors_text":"Lukas Liehr, Mitchell A. Taylor, Peiyang Yu","submitted_at":"2026-08-12T17:23:03Z","abstract_excerpt":"We establish a sharp version of Grothendieck's theorem for Bessel sequences. Precisely, given a Bessel sequence $\\{ x_j \\}_{j\\in\\mathbb{N}}$ with Bessel bound $1$ in a Hilbert space, we show that there exists functions $\\{ f_j \\}_{j\\in\\mathbb{N}}$ belonging to the unit ball of $L^\\infty([0,1])$ such that for all $j,k \\in \\mathbb{N}$ one has $$ \\langle x_j,x_k\\rangle = \\int_0^1 f_j(x)\\overline{f_k(x)}\\,dx.$$ As an application, we give an affirmative answer to an extension problem of Olevskii: if $E \\subset [0,1]$ is a Lebesgue measurable set such that $[0,1]\\setminus E$ has positive measure, th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.12280","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2608.12280/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}