{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:LDANYEYO4LBBLLAI5MPYWBAVCP","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"d1eb6b0e28a4e8b2c8903e1899022de01f0f0649f8f56b76fdddd52ca33ba6cb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-06-20T17:50:10Z","title_canon_sha256":"ae8b16ecf6a0725aa28ac9d8ac93629270e547ce66b257f06c6479f15489825a"},"schema_version":"1.0","source":{"id":"1806.07866","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.07866","created_at":"2026-05-18T00:09:41Z"},{"alias_kind":"arxiv_version","alias_value":"1806.07866v2","created_at":"2026-05-18T00:09:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.07866","created_at":"2026-05-18T00:09:41Z"},{"alias_kind":"pith_short_12","alias_value":"LDANYEYO4LBB","created_at":"2026-05-18T12:32:37Z"},{"alias_kind":"pith_short_16","alias_value":"LDANYEYO4LBBLLAI","created_at":"2026-05-18T12:32:37Z"},{"alias_kind":"pith_short_8","alias_value":"LDANYEYO","created_at":"2026-05-18T12:32:37Z"}],"graph_snapshots":[{"event_id":"sha256:82818f5df600c9461e9acd9e592f7d60f11daf1eb1e528e4b6b8bbec198f1811","target":"graph","created_at":"2026-05-18T00:09:41Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $\\mathcal{H}$ be a complex separable Hilbert space. We prove that if $\\{f_{n}\\}_{n=1}^{\\infty}$ is a Schauder basis of the Hilbert space $\\mathcal{H}$, then the angles between any two vectors in this basis must have a positive lower bound. Furthermore, we investigate that $\\{z^{\\sigma^{-1}(n)}\\}_{n=1}^{\\infty}$ can never be a Schauder basis of $L^{2}(\\mathbb{T},\\nu)$, where $\\mathbb{T}$ is the unit circle, $\\nu$ is a finite positive discrete measure, and $\\sigma: \\mathbb{Z} \\rightarrow \\mathbb{N}$ is an arbitrary surjective and injective map.","authors_text":"Bingzhe Hou, Geng Tian, Xinzhi Zhang, Yang Cao","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-06-20T17:50:10Z","title":"Angles and Schauder basis in Hilbert spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.07866","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:599a29b5326d28001a0bda29f0fb8167da36513eef65947f672bbb74eb8bb240","target":"record","created_at":"2026-05-18T00:09:41Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"d1eb6b0e28a4e8b2c8903e1899022de01f0f0649f8f56b76fdddd52ca33ba6cb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-06-20T17:50:10Z","title_canon_sha256":"ae8b16ecf6a0725aa28ac9d8ac93629270e547ce66b257f06c6479f15489825a"},"schema_version":"1.0","source":{"id":"1806.07866","kind":"arxiv","version":2}},"canonical_sha256":"58c0dc130ee2c215ac08eb1f8b041513c7cb9573116711931588b4a0860cb442","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"58c0dc130ee2c215ac08eb1f8b041513c7cb9573116711931588b4a0860cb442","first_computed_at":"2026-05-18T00:09:41.354525Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:09:41.354525Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9lM/AEpIKVyIkgi8eF7quHpxe4w9B9LCkgdhDg8+GoZTCA0ccEySuQbFvIpWLAjYNT4ndpEEMbJMdtH78oylAA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:09:41.355027Z","signed_message":"canonical_sha256_bytes"},"source_id":"1806.07866","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:599a29b5326d28001a0bda29f0fb8167da36513eef65947f672bbb74eb8bb240","sha256:82818f5df600c9461e9acd9e592f7d60f11daf1eb1e528e4b6b8bbec198f1811"],"state_sha256":"65cda51eff839802fe79045dd94bc3e6fcf5672c62f52f4dde6325f1676c25ee"}