{"paper":{"title":"Pre-Hilbert spaces without orthonormal bases","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Saka\\'e Fuchino","submitted_at":"2016-06-13T09:26:49Z","abstract_excerpt":"We give an algebraic characterization of pre-Hilbert spaces with an orthonormal basis. This characterization is used to show that there are pre-Hilbert spaces $X$ of dimension and density $\\lambda$ for any uncountable $\\lambda$ without any orthonormal basis.\n  Let us call a pre-Hilbert space without any orthonormal bases pathological. The pair of the cardinals $\\kappa\\leq\\lambda$ such that there is a pre-Hilbert space of dimension $\\kappa$ and density $\\lambda$ are known to be characterized by the inequality $\\lambda\\leq\\kappa^{\\aleph_0}$. Our result implies that there are pathological pre-Hil"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.03869","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"}