{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:CW3HBTAKGUN4RVBRJLNVOOXCSD","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":"82b0235f70ce5267b9885d2a73355caf43e9bd33f0f4a35c37cdea412cd76c3e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-07-31T11:57:09Z","title_canon_sha256":"7a93e78e3880b63f1843636804ae417ebe251e4e265f0c60fcf6e8402f647d56"},"schema_version":"1.0","source":{"id":"1108.0169","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1108.0169","created_at":"2026-05-18T04:05:57Z"},{"alias_kind":"arxiv_version","alias_value":"1108.0169v3","created_at":"2026-05-18T04:05:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.0169","created_at":"2026-05-18T04:05:57Z"},{"alias_kind":"pith_short_12","alias_value":"CW3HBTAKGUN4","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_16","alias_value":"CW3HBTAKGUN4RVBR","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_8","alias_value":"CW3HBTAK","created_at":"2026-05-18T12:26:26Z"}],"graph_snapshots":[{"event_id":"sha256:52905c6a3a3f3778b231c0de8751edd01420f99bf48dff1d1e6560bb7faa4b36","target":"graph","created_at":"2026-05-18T04:05:57Z","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":"We prove a criterion for continuity of bilinear maps on countable direct sums of topological vector spaces. As a first application, we get a new proof for the fact (due to Hirai et al. 2001) that the map taking a pair of test functions on R^n to their convolution is continuous. The criterion also allows an open problem by K.-H. Neeb to be solved: If E is a locally convex space, regard the tensor algebra T(E) as the locally convex direct sum of the projective tensor powers T^j(E) of E. We show that T(E) is a topological algebra if and only if every sequence of continuous seminorms on E has an u","authors_text":"Helge Glockner","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-07-31T11:57:09Z","title":"Continuity of bilinear maps on direct sums of topological vector spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.0169","kind":"arxiv","version":3},"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":"sha256:73479b85dd9ecdf165eb1cb66b6935f7fba71210392645ce4fbb7f4ca3c13c54","target":"record","created_at":"2026-05-18T04:05:57Z","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":"82b0235f70ce5267b9885d2a73355caf43e9bd33f0f4a35c37cdea412cd76c3e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-07-31T11:57:09Z","title_canon_sha256":"7a93e78e3880b63f1843636804ae417ebe251e4e265f0c60fcf6e8402f647d56"},"schema_version":"1.0","source":{"id":"1108.0169","kind":"arxiv","version":3}},"canonical_sha256":"15b670cc0a351bc8d4314adb573ae290dae833bd4ba0a9b9d3930b47f153b257","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"15b670cc0a351bc8d4314adb573ae290dae833bd4ba0a9b9d3930b47f153b257","first_computed_at":"2026-05-18T04:05:57.548355Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:05:57.548355Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CjWTOV5hYX8D3OY6DPzu4mSBURajQfWYglL9Gtztf8Z4ikOpwoA9qj/JJsYFtxiHTiUY8B3ExhkW2hMJUZxnDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:05:57.549171Z","signed_message":"canonical_sha256_bytes"},"source_id":"1108.0169","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:73479b85dd9ecdf165eb1cb66b6935f7fba71210392645ce4fbb7f4ca3c13c54","sha256:52905c6a3a3f3778b231c0de8751edd01420f99bf48dff1d1e6560bb7faa4b36"],"state_sha256":"f2b0fc64f4bcf5de8caf91f8e227eea2f6c73de9ff00f9a3a3f87302bba2f037"}