{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:4CSMNWHRWFVHAMY3V7TVCZHW7G","short_pith_number":"pith:4CSMNWHR","canonical_record":{"source":{"id":"1610.00423","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-10-03T06:45:10Z","cross_cats_sorted":[],"title_canon_sha256":"ab6103004d231f849ce3b3d30ed2285a8b94e69f0e8a7176795c0f40f76f474c","abstract_canon_sha256":"17182e1e74fa11650509bf01766f1a707a759f670af7c671bc3f77bdb9c7004d"},"schema_version":"1.0"},"canonical_sha256":"e0a4c6d8f1b16a70331bafe75164f6f99049e20a8c29b8ed69c0e8d34c1335dc","source":{"kind":"arxiv","id":"1610.00423","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1610.00423","created_at":"2026-05-18T00:53:39Z"},{"alias_kind":"arxiv_version","alias_value":"1610.00423v2","created_at":"2026-05-18T00:53:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.00423","created_at":"2026-05-18T00:53:39Z"},{"alias_kind":"pith_short_12","alias_value":"4CSMNWHRWFVH","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_16","alias_value":"4CSMNWHRWFVHAMY3","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_8","alias_value":"4CSMNWHR","created_at":"2026-05-18T12:29:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:4CSMNWHRWFVHAMY3V7TVCZHW7G","target":"record","payload":{"canonical_record":{"source":{"id":"1610.00423","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-10-03T06:45:10Z","cross_cats_sorted":[],"title_canon_sha256":"ab6103004d231f849ce3b3d30ed2285a8b94e69f0e8a7176795c0f40f76f474c","abstract_canon_sha256":"17182e1e74fa11650509bf01766f1a707a759f670af7c671bc3f77bdb9c7004d"},"schema_version":"1.0"},"canonical_sha256":"e0a4c6d8f1b16a70331bafe75164f6f99049e20a8c29b8ed69c0e8d34c1335dc","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:53:39.944231Z","signature_b64":"8JDcTFghb9q/MymLA5mh8mdtXiwpaeRzqUG2oZERu5Z8bSRcxHSd4sKKNrMb73PBuzTymiTIiGbSuzTflF7VDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e0a4c6d8f1b16a70331bafe75164f6f99049e20a8c29b8ed69c0e8d34c1335dc","last_reissued_at":"2026-05-18T00:53:39.943722Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:53:39.943722Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1610.00423","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:53:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hCgzdePqPHettDmqJl2Hh3mi3uwvqQ/eKsP98YzWeiYilhwAbr8Y/2y6lPys7X66IMCLvDhZQ8L/DDEiOwuHCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T10:50:45.810629Z"},"content_sha256":"4309430ad646426cd170129060b6db63b9c8f43bdc25abfa04b5bda643ab127e","schema_version":"1.0","event_id":"sha256:4309430ad646426cd170129060b6db63b9c8f43bdc25abfa04b5bda643ab127e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:4CSMNWHRWFVHAMY3V7TVCZHW7G","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Decomposition of functions between Banach spaces in the orthogonality equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Maysam Maysami Sadr","submitted_at":"2016-10-03T06:45:10Z","abstract_excerpt":"Let $E,F$ be Banach spaces. In the case that $F$ is reflexive we give a description for the solutions $(f,g)$ of the Banach-orthogonality equation $$\\langle f(x),g(\\alpha)\\rangle=\\langle x,\\alpha\\rangle\\hspace{10mm}\\forall x\\in E,\\forall \\alpha\\in E^*,$$ where $f:E\\rightarrow F,g:E^*\\rightarrow F^*$ are two maps. Our result generalizes the recent result of {\\L}ukasik and W\\'{o}jcik in the case that $E$ and $F$ are Hilbert spaces."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.00423","kind":"arxiv","version":2},"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":"2026-05-18T00:53:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tVSAVb8GWntoB1OSugjnaJZWI94tFnGFu3aY7o7dEubdQdJijn7VTyfXXnSOkpdOveBn4fqpReJCTt0X13NcBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T10:50:45.811152Z"},"content_sha256":"5934de07034f42361ff06f7b831f1ba3637646d8d6955221a57c0d7d6e3ab006","schema_version":"1.0","event_id":"sha256:5934de07034f42361ff06f7b831f1ba3637646d8d6955221a57c0d7d6e3ab006"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4CSMNWHRWFVHAMY3V7TVCZHW7G/bundle.json","state_url":"https://pith.science/pith/4CSMNWHRWFVHAMY3V7TVCZHW7G/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4CSMNWHRWFVHAMY3V7TVCZHW7G/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":"2026-06-25T10:50:45Z","links":{"resolver":"https://pith.science/pith/4CSMNWHRWFVHAMY3V7TVCZHW7G","bundle":"https://pith.science/pith/4CSMNWHRWFVHAMY3V7TVCZHW7G/bundle.json","state":"https://pith.science/pith/4CSMNWHRWFVHAMY3V7TVCZHW7G/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4CSMNWHRWFVHAMY3V7TVCZHW7G/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:4CSMNWHRWFVHAMY3V7TVCZHW7G","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":"17182e1e74fa11650509bf01766f1a707a759f670af7c671bc3f77bdb9c7004d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-10-03T06:45:10Z","title_canon_sha256":"ab6103004d231f849ce3b3d30ed2285a8b94e69f0e8a7176795c0f40f76f474c"},"schema_version":"1.0","source":{"id":"1610.00423","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1610.00423","created_at":"2026-05-18T00:53:39Z"},{"alias_kind":"arxiv_version","alias_value":"1610.00423v2","created_at":"2026-05-18T00:53:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.00423","created_at":"2026-05-18T00:53:39Z"},{"alias_kind":"pith_short_12","alias_value":"4CSMNWHRWFVH","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_16","alias_value":"4CSMNWHRWFVHAMY3","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_8","alias_value":"4CSMNWHR","created_at":"2026-05-18T12:29:58Z"}],"graph_snapshots":[{"event_id":"sha256:5934de07034f42361ff06f7b831f1ba3637646d8d6955221a57c0d7d6e3ab006","target":"graph","created_at":"2026-05-18T00:53:39Z","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 $E,F$ be Banach spaces. In the case that $F$ is reflexive we give a description for the solutions $(f,g)$ of the Banach-orthogonality equation $$\\langle f(x),g(\\alpha)\\rangle=\\langle x,\\alpha\\rangle\\hspace{10mm}\\forall x\\in E,\\forall \\alpha\\in E^*,$$ where $f:E\\rightarrow F,g:E^*\\rightarrow F^*$ are two maps. Our result generalizes the recent result of {\\L}ukasik and W\\'{o}jcik in the case that $E$ and $F$ are Hilbert spaces.","authors_text":"Maysam Maysami Sadr","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-10-03T06:45:10Z","title":"Decomposition of functions between Banach spaces in the orthogonality equation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.00423","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":"sha256:4309430ad646426cd170129060b6db63b9c8f43bdc25abfa04b5bda643ab127e","target":"record","created_at":"2026-05-18T00:53:39Z","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":"17182e1e74fa11650509bf01766f1a707a759f670af7c671bc3f77bdb9c7004d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-10-03T06:45:10Z","title_canon_sha256":"ab6103004d231f849ce3b3d30ed2285a8b94e69f0e8a7176795c0f40f76f474c"},"schema_version":"1.0","source":{"id":"1610.00423","kind":"arxiv","version":2}},"canonical_sha256":"e0a4c6d8f1b16a70331bafe75164f6f99049e20a8c29b8ed69c0e8d34c1335dc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e0a4c6d8f1b16a70331bafe75164f6f99049e20a8c29b8ed69c0e8d34c1335dc","first_computed_at":"2026-05-18T00:53:39.943722Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:53:39.943722Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8JDcTFghb9q/MymLA5mh8mdtXiwpaeRzqUG2oZERu5Z8bSRcxHSd4sKKNrMb73PBuzTymiTIiGbSuzTflF7VDA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:53:39.944231Z","signed_message":"canonical_sha256_bytes"},"source_id":"1610.00423","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4309430ad646426cd170129060b6db63b9c8f43bdc25abfa04b5bda643ab127e","sha256:5934de07034f42361ff06f7b831f1ba3637646d8d6955221a57c0d7d6e3ab006"],"state_sha256":"21da56b41fc92afcc5b87420660f1c05badee17217ea8307707039762db9cf7a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Sp1HNCo/SZA3tGuT8qEWLN0Iw68TdDp3L+wRMLpO1LIoJ2VgXZmzrfNKxZ3EL1dCaOL+uwNtp8jlp4tU6K1ICQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T10:50:45.813157Z","bundle_sha256":"3747f46d70fea080af07ee44834ff8513fbe652a2e10b1895e6170bbb2f32ad6"}}