{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:1994:U66EQ5FERHTPERJW5ICBFDAQ3S","short_pith_number":"pith:U66EQ5FE","canonical_record":{"source":{"id":"math/9402210","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.FA","submitted_at":"1994-02-17T18:28:30Z","cross_cats_sorted":[],"title_canon_sha256":"83a4d44b1412f64b8944219af40173ba3d208c7c0f5784307fbef5179c999f4a","abstract_canon_sha256":"34e5cbbda6d275f6a477720511c8174ef7849bffba3dbe53e8a265c53353731d"},"schema_version":"1.0"},"canonical_sha256":"a7bc4874a489e6f24536ea04128c10dca8ab6f0229a6f6153548050c4f036149","source":{"kind":"arxiv","id":"math/9402210","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9402210","created_at":"2026-05-18T01:05:51Z"},{"alias_kind":"arxiv_version","alias_value":"math/9402210v1","created_at":"2026-05-18T01:05:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9402210","created_at":"2026-05-18T01:05:51Z"},{"alias_kind":"pith_short_12","alias_value":"U66EQ5FERHTP","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_16","alias_value":"U66EQ5FERHTPERJW","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_8","alias_value":"U66EQ5FE","created_at":"2026-05-18T12:25:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:1994:U66EQ5FERHTPERJW5ICBFDAQ3S","target":"record","payload":{"canonical_record":{"source":{"id":"math/9402210","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.FA","submitted_at":"1994-02-17T18:28:30Z","cross_cats_sorted":[],"title_canon_sha256":"83a4d44b1412f64b8944219af40173ba3d208c7c0f5784307fbef5179c999f4a","abstract_canon_sha256":"34e5cbbda6d275f6a477720511c8174ef7849bffba3dbe53e8a265c53353731d"},"schema_version":"1.0"},"canonical_sha256":"a7bc4874a489e6f24536ea04128c10dca8ab6f0229a6f6153548050c4f036149","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:51.649271Z","signature_b64":"nSjASkNkEjo1DlCQKbKvYVwPie32g/z3CP7dnUZwwv3M5XVdspZ3L9E2qWU+JezPsqCQsHXIlQVePLc1Anr/Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a7bc4874a489e6f24536ea04128c10dca8ab6f0229a6f6153548050c4f036149","last_reissued_at":"2026-05-18T01:05:51.648831Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:51.648831Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/9402210","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:05:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jaHM3RSUYRC7BdsLYlFjADnxrFCbH32m1a50XUM5cXeZ/vFVfb+XvlfcledQGg3pFSLNN5PGF2joUWOvrYLrCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T06:28:53.235696Z"},"content_sha256":"e9c68e880871188fe70f631ff12cd526356fccf3adb6f290e0cfdec573278dcd","schema_version":"1.0","event_id":"sha256:e9c68e880871188fe70f631ff12cd526356fccf3adb6f290e0cfdec573278dcd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:1994:U66EQ5FERHTPERJW5ICBFDAQ3S","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"From weak to strong types of $L_E^1$-convergence by the Bocce-criterion","license":"","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Erik J. Balder, Maria Girardi, Vincent Jalby","submitted_at":"1994-02-17T18:28:30Z","abstract_excerpt":"Necessary and sufficient oscillation conditions are given for a weakly convergent sequence (resp. relatively weakly compact set) in the Bochner-Lebesgue space $\\l1$ to be norm convergent (resp. relatively norm compact), thus extending the known results for $\\rl1$. Similarly, necessary and sufficient oscillation conditions are given to pass from weak to limited (and also to Pettis-norm) convergence in $\\l1$. It is shown that tightness is a necessary and sufficient condition to pass from limited to strong convergence. Other implications between several modes of convergence in $\\l1$ are also stud"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9402210","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:05:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FxawrZhvUUSdsIQiqMxki1Hf+KTBWFkwT/oQUtE2tdpV5kf5+OUzYUtewdtAAlc3n3tv/I222y0ZuNI9C0buCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T06:28:53.236263Z"},"content_sha256":"a7110d72e03c894269193d3140d18a309322f55ef128fda8cb7808b9aa5f090b","schema_version":"1.0","event_id":"sha256:a7110d72e03c894269193d3140d18a309322f55ef128fda8cb7808b9aa5f090b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/U66EQ5FERHTPERJW5ICBFDAQ3S/bundle.json","state_url":"https://pith.science/pith/U66EQ5FERHTPERJW5ICBFDAQ3S/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/U66EQ5FERHTPERJW5ICBFDAQ3S/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-05T06:28:53Z","links":{"resolver":"https://pith.science/pith/U66EQ5FERHTPERJW5ICBFDAQ3S","bundle":"https://pith.science/pith/U66EQ5FERHTPERJW5ICBFDAQ3S/bundle.json","state":"https://pith.science/pith/U66EQ5FERHTPERJW5ICBFDAQ3S/state.json","well_known_bundle":"https://pith.science/.well-known/pith/U66EQ5FERHTPERJW5ICBFDAQ3S/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:1994:U66EQ5FERHTPERJW5ICBFDAQ3S","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":"34e5cbbda6d275f6a477720511c8174ef7849bffba3dbe53e8a265c53353731d","cross_cats_sorted":[],"license":"","primary_cat":"math.FA","submitted_at":"1994-02-17T18:28:30Z","title_canon_sha256":"83a4d44b1412f64b8944219af40173ba3d208c7c0f5784307fbef5179c999f4a"},"schema_version":"1.0","source":{"id":"math/9402210","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9402210","created_at":"2026-05-18T01:05:51Z"},{"alias_kind":"arxiv_version","alias_value":"math/9402210v1","created_at":"2026-05-18T01:05:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9402210","created_at":"2026-05-18T01:05:51Z"},{"alias_kind":"pith_short_12","alias_value":"U66EQ5FERHTP","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_16","alias_value":"U66EQ5FERHTPERJW","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_8","alias_value":"U66EQ5FE","created_at":"2026-05-18T12:25:47Z"}],"graph_snapshots":[{"event_id":"sha256:a7110d72e03c894269193d3140d18a309322f55ef128fda8cb7808b9aa5f090b","target":"graph","created_at":"2026-05-18T01:05:51Z","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":"Necessary and sufficient oscillation conditions are given for a weakly convergent sequence (resp. relatively weakly compact set) in the Bochner-Lebesgue space $\\l1$ to be norm convergent (resp. relatively norm compact), thus extending the known results for $\\rl1$. Similarly, necessary and sufficient oscillation conditions are given to pass from weak to limited (and also to Pettis-norm) convergence in $\\l1$. It is shown that tightness is a necessary and sufficient condition to pass from limited to strong convergence. Other implications between several modes of convergence in $\\l1$ are also stud","authors_text":"Erik J. Balder, Maria Girardi, Vincent Jalby","cross_cats":[],"headline":"","license":"","primary_cat":"math.FA","submitted_at":"1994-02-17T18:28:30Z","title":"From weak to strong types of $L_E^1$-convergence by the Bocce-criterion"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9402210","kind":"arxiv","version":1},"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:e9c68e880871188fe70f631ff12cd526356fccf3adb6f290e0cfdec573278dcd","target":"record","created_at":"2026-05-18T01:05:51Z","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":"34e5cbbda6d275f6a477720511c8174ef7849bffba3dbe53e8a265c53353731d","cross_cats_sorted":[],"license":"","primary_cat":"math.FA","submitted_at":"1994-02-17T18:28:30Z","title_canon_sha256":"83a4d44b1412f64b8944219af40173ba3d208c7c0f5784307fbef5179c999f4a"},"schema_version":"1.0","source":{"id":"math/9402210","kind":"arxiv","version":1}},"canonical_sha256":"a7bc4874a489e6f24536ea04128c10dca8ab6f0229a6f6153548050c4f036149","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a7bc4874a489e6f24536ea04128c10dca8ab6f0229a6f6153548050c4f036149","first_computed_at":"2026-05-18T01:05:51.648831Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:05:51.648831Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"nSjASkNkEjo1DlCQKbKvYVwPie32g/z3CP7dnUZwwv3M5XVdspZ3L9E2qWU+JezPsqCQsHXIlQVePLc1Anr/Dw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:05:51.649271Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/9402210","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e9c68e880871188fe70f631ff12cd526356fccf3adb6f290e0cfdec573278dcd","sha256:a7110d72e03c894269193d3140d18a309322f55ef128fda8cb7808b9aa5f090b"],"state_sha256":"81faf17d1d991d8e4ffb13996936b584d16b111f0e108bb3bf881f9e55a43aa8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3yDDAD8xiOOuN0qs24YC9bQpCsbsEjt9bNQSIiNIcb0zQQ2WfYNrSfLQV0c10lTRQc5mVeJy9zjh5KTSLqlmDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T06:28:53.240032Z","bundle_sha256":"1ce27525b727a72efe15376afc31f808f4d2311ac559a0534f151a064c5dcc19"}}