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The main theorem is to show that the following conditions are equivalent:\n  (1) the unit vector basis is boundedly complete;\n  (2) $L$ is $F_\\sigma$ in $\\mathbb R^\\mathbb N$;\n  (3) $\\mathbb R^\\mathbb N/L$ is Borel reducible to $\\mathbb R^\\mathbb N/\\ell_\\infty$.\n  We show that any Schauder equivalence relation generalized by basis of $\\ell_2$ is Borel bireducible to $\\mathbb R^\\mathbb N/\\ell_2$ "},"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":"1504.00299","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-04-01T17:16:11Z","cross_cats_sorted":[],"title_canon_sha256":"d13c1701a1e14069b5d728ebf1af4b07343858bd9cde503ad172132e8c61324c","abstract_canon_sha256":"93218537c28c66fdae25e85eac698e33735e115173a866cb21500c92db1bf1ae"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:19:44.125847Z","signature_b64":"eKvfZfI+XMUgJdsOho48loSDm7wts1z6h0vyzO9sXOmx7i4aY3TK8dKEODYV2VqoBnFFqkIVcBbVPR8pdgtzAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cfb0dec85715cb2fd11a2c4b39523b270986099d1a43efb07e960684e104d677","last_reissued_at":"2026-05-18T02:19:44.125423Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:19:44.125423Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On equivalence relations generated by Schauder bases","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Longyun Ding","submitted_at":"2015-04-01T17:16:11Z","abstract_excerpt":"In this paper, a notion of Schauder equivalence relation $\\mathbb R^\\mathbb N/L$ is introduced, where $L$ is a linear subspace of $\\mathbb R^\\mathbb N$ and the unit vectors form a Schauder basis of $L$. The main theorem is to show that the following conditions are equivalent:\n  (1) the unit vector basis is boundedly complete;\n  (2) $L$ is $F_\\sigma$ in $\\mathbb R^\\mathbb N$;\n  (3) $\\mathbb R^\\mathbb N/L$ is Borel reducible to $\\mathbb R^\\mathbb N/\\ell_\\infty$.\n  We show that any Schauder equivalence relation generalized by basis of $\\ell_2$ is Borel bireducible to $\\mathbb R^\\mathbb N/\\ell_2$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.00299","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1504.00299","created_at":"2026-05-18T02:19:44.125486+00:00"},{"alias_kind":"arxiv_version","alias_value":"1504.00299v1","created_at":"2026-05-18T02:19:44.125486+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.00299","created_at":"2026-05-18T02:19:44.125486+00:00"},{"alias_kind":"pith_short_12","alias_value":"Z6YN5SCXCXFS","created_at":"2026-05-18T12:29:52.810259+00:00"},{"alias_kind":"pith_short_16","alias_value":"Z6YN5SCXCXFS7UI2","created_at":"2026-05-18T12:29:52.810259+00:00"},{"alias_kind":"pith_short_8","alias_value":"Z6YN5SCX","created_at":"2026-05-18T12:29:52.810259+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/Z6YN5SCXCXFS7UI2FRFTSUR3E4","json":"https://pith.science/pith/Z6YN5SCXCXFS7UI2FRFTSUR3E4.json","graph_json":"https://pith.science/api/pith-number/Z6YN5SCXCXFS7UI2FRFTSUR3E4/graph.json","events_json":"https://pith.science/api/pith-number/Z6YN5SCXCXFS7UI2FRFTSUR3E4/events.json","paper":"https://pith.science/paper/Z6YN5SCX"},"agent_actions":{"view_html":"https://pith.science/pith/Z6YN5SCXCXFS7UI2FRFTSUR3E4","download_json":"https://pith.science/pith/Z6YN5SCXCXFS7UI2FRFTSUR3E4.json","view_paper":"https://pith.science/paper/Z6YN5SCX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1504.00299&json=true","fetch_graph":"https://pith.science/api/pith-number/Z6YN5SCXCXFS7UI2FRFTSUR3E4/graph.json","fetch_events":"https://pith.science/api/pith-number/Z6YN5SCXCXFS7UI2FRFTSUR3E4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/Z6YN5SCXCXFS7UI2FRFTSUR3E4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/Z6YN5SCXCXFS7UI2FRFTSUR3E4/action/storage_attestation","attest_author":"https://pith.science/pith/Z6YN5SCXCXFS7UI2FRFTSUR3E4/action/author_attestation","sign_citation":"https://pith.science/pith/Z6YN5SCXCXFS7UI2FRFTSUR3E4/action/citation_signature","submit_replication":"https://pith.science/pith/Z6YN5SCXCXFS7UI2FRFTSUR3E4/action/replication_record"}},"created_at":"2026-05-18T02:19:44.125486+00:00","updated_at":"2026-05-18T02:19:44.125486+00:00"}