{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:1993:IDJKVJG5WRW3R6OXJLDLFT75TV","short_pith_number":"pith:IDJKVJG5","canonical_record":{"source":{"id":"math/9303203","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.FA","submitted_at":"1993-03-29T17:51:57Z","cross_cats_sorted":[],"title_canon_sha256":"5af923bf8f3be959c92751af8dd611238444a13e8896c8027e19331ef5dbdaa0","abstract_canon_sha256":"d946e0fc2547b14505c5ceed89935472438d5e0f5ccc932333fda8c78f13d412"},"schema_version":"1.0"},"canonical_sha256":"40d2aaa4ddb46db8f9d74ac6b2cffd9d7f85d55715f3518f3e006d40aefcd951","source":{"kind":"arxiv","id":"math/9303203","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9303203","created_at":"2026-05-18T04:41:16Z"},{"alias_kind":"arxiv_version","alias_value":"math/9303203v1","created_at":"2026-05-18T04:41:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9303203","created_at":"2026-05-18T04:41:16Z"},{"alias_kind":"pith_short_12","alias_value":"IDJKVJG5WRW3","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_16","alias_value":"IDJKVJG5WRW3R6OX","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_8","alias_value":"IDJKVJG5","created_at":"2026-05-18T12:25:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:1993:IDJKVJG5WRW3R6OXJLDLFT75TV","target":"record","payload":{"canonical_record":{"source":{"id":"math/9303203","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.FA","submitted_at":"1993-03-29T17:51:57Z","cross_cats_sorted":[],"title_canon_sha256":"5af923bf8f3be959c92751af8dd611238444a13e8896c8027e19331ef5dbdaa0","abstract_canon_sha256":"d946e0fc2547b14505c5ceed89935472438d5e0f5ccc932333fda8c78f13d412"},"schema_version":"1.0"},"canonical_sha256":"40d2aaa4ddb46db8f9d74ac6b2cffd9d7f85d55715f3518f3e006d40aefcd951","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:41:16.668209Z","signature_b64":"4j5VrXn4XHxQFOhQISUIctFiOPFNVtH50mVNHoSXB6h5tV6E3BIG+W4fgC2vfYe5SkXAyhUhPtNbmDuVaYPjAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"40d2aaa4ddb46db8f9d74ac6b2cffd9d7f85d55715f3518f3e006d40aefcd951","last_reissued_at":"2026-05-18T04:41:16.667635Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:41:16.667635Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/9303203","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":"2026-05-18T04:41:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8Qq4c4hTtPIAbRzfNySqFeJL/9Fr7c+gxIfKB4C0/kHaaUCH0OqGfazg5OFRQkexeIFTzfUIyPN1B1KDem4ZCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T02:00:59.989991Z"},"content_sha256":"b0fa0ed467ab39b670d3d037c14c3ebf360dcb8d9eef371c8b36b5b847a73648","schema_version":"1.0","event_id":"sha256:b0fa0ed467ab39b670d3d037c14c3ebf360dcb8d9eef371c8b36b5b847a73648"},{"event_type":"graph_snapshot","subject_pith_number":"pith:1993:IDJKVJG5WRW3R6OXJLDLFT75TV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"W^*-derived sets of transfinite order of subspaces of dual Banach spaces","license":"","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Mikhail I. Ostrovskii","submitted_at":"1993-03-29T17:51:57Z","abstract_excerpt":"It is an English translation of the paper originally published in Russian and Ukrainian in 1987. In the appendix of his book S.Banach introduced the following definition Let $X$ be a Banach space and $\\Gamma$ be a subspace of the dual space $X^*$. The set of all limits of $w^{*}$-convergent sequences in $\\Gamma $ is called the $w^*${\\it -derived set} of $\\Gamma $ and is denoted by $\\Gamma _{(1)}$. For an ordinal $\\alpha$ the $w^{*}$-{\\it derived set of order} $\\alpha $ is defined inductively by the equality: $$ \\Gamma _{(\\alpha )}=\\bigcup _{\\beta <\\alpha }((\\Gamma _{(\\beta )})_{(1)}. $$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9303203","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":"2026-05-18T04:41:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jr0b7sBSrYicvbNyT3kr0DbRwi1XROCFP2q3UkCS1pfY+0Z+MP8hJlMgVZ504DJoE+xMUpuC8Xnk0Wst298fAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T02:00:59.990344Z"},"content_sha256":"a54289745f34c587e76f2d9f7cc4a899c781e5398da2577093bbe7881bb954a8","schema_version":"1.0","event_id":"sha256:a54289745f34c587e76f2d9f7cc4a899c781e5398da2577093bbe7881bb954a8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/IDJKVJG5WRW3R6OXJLDLFT75TV/bundle.json","state_url":"https://pith.science/pith/IDJKVJG5WRW3R6OXJLDLFT75TV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/IDJKVJG5WRW3R6OXJLDLFT75TV/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":"2026-06-03T02:00:59Z","links":{"resolver":"https://pith.science/pith/IDJKVJG5WRW3R6OXJLDLFT75TV","bundle":"https://pith.science/pith/IDJKVJG5WRW3R6OXJLDLFT75TV/bundle.json","state":"https://pith.science/pith/IDJKVJG5WRW3R6OXJLDLFT75TV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/IDJKVJG5WRW3R6OXJLDLFT75TV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:1993:IDJKVJG5WRW3R6OXJLDLFT75TV","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":"d946e0fc2547b14505c5ceed89935472438d5e0f5ccc932333fda8c78f13d412","cross_cats_sorted":[],"license":"","primary_cat":"math.FA","submitted_at":"1993-03-29T17:51:57Z","title_canon_sha256":"5af923bf8f3be959c92751af8dd611238444a13e8896c8027e19331ef5dbdaa0"},"schema_version":"1.0","source":{"id":"math/9303203","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9303203","created_at":"2026-05-18T04:41:16Z"},{"alias_kind":"arxiv_version","alias_value":"math/9303203v1","created_at":"2026-05-18T04:41:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9303203","created_at":"2026-05-18T04:41:16Z"},{"alias_kind":"pith_short_12","alias_value":"IDJKVJG5WRW3","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_16","alias_value":"IDJKVJG5WRW3R6OX","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_8","alias_value":"IDJKVJG5","created_at":"2026-05-18T12:25:47Z"}],"graph_snapshots":[{"event_id":"sha256:a54289745f34c587e76f2d9f7cc4a899c781e5398da2577093bbe7881bb954a8","target":"graph","created_at":"2026-05-18T04:41:16Z","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":"It is an English translation of the paper originally published in Russian and Ukrainian in 1987. In the appendix of his book S.Banach introduced the following definition Let $X$ be a Banach space and $\\Gamma$ be a subspace of the dual space $X^*$. The set of all limits of $w^{*}$-convergent sequences in $\\Gamma $ is called the $w^*${\\it -derived set} of $\\Gamma $ and is denoted by $\\Gamma _{(1)}$. For an ordinal $\\alpha$ the $w^{*}$-{\\it derived set of order} $\\alpha $ is defined inductively by the equality: $$ \\Gamma _{(\\alpha )}=\\bigcup _{\\beta <\\alpha }((\\Gamma _{(\\beta )})_{(1)}. $$","authors_text":"Mikhail I. Ostrovskii","cross_cats":[],"headline":"","license":"","primary_cat":"math.FA","submitted_at":"1993-03-29T17:51:57Z","title":"W^*-derived sets of transfinite order of subspaces of dual Banach spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9303203","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:b0fa0ed467ab39b670d3d037c14c3ebf360dcb8d9eef371c8b36b5b847a73648","target":"record","created_at":"2026-05-18T04:41:16Z","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":"d946e0fc2547b14505c5ceed89935472438d5e0f5ccc932333fda8c78f13d412","cross_cats_sorted":[],"license":"","primary_cat":"math.FA","submitted_at":"1993-03-29T17:51:57Z","title_canon_sha256":"5af923bf8f3be959c92751af8dd611238444a13e8896c8027e19331ef5dbdaa0"},"schema_version":"1.0","source":{"id":"math/9303203","kind":"arxiv","version":1}},"canonical_sha256":"40d2aaa4ddb46db8f9d74ac6b2cffd9d7f85d55715f3518f3e006d40aefcd951","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"40d2aaa4ddb46db8f9d74ac6b2cffd9d7f85d55715f3518f3e006d40aefcd951","first_computed_at":"2026-05-18T04:41:16.667635Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:41:16.667635Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4j5VrXn4XHxQFOhQISUIctFiOPFNVtH50mVNHoSXB6h5tV6E3BIG+W4fgC2vfYe5SkXAyhUhPtNbmDuVaYPjAA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:41:16.668209Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/9303203","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b0fa0ed467ab39b670d3d037c14c3ebf360dcb8d9eef371c8b36b5b847a73648","sha256:a54289745f34c587e76f2d9f7cc4a899c781e5398da2577093bbe7881bb954a8"],"state_sha256":"39331d0120845951f6b5ab834b7c1f6765789e9c091f56663d63020b0f9e9113"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Mpa4aCpB6pvsvJ5UD1MljbPro4QODpCmilitpFr7MiN5z8om+ya9jY9+x1bGn3pEJ8i2HiVQkVItbZun01aBDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T02:00:59.992384Z","bundle_sha256":"f98ddcfad28b1bf10707ce7cd01f73f5bb8714a7765a4ead91b9bc4ca4a20116"}}