{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:MZHIGPL7WDUHWU343NNYXYIHXL","short_pith_number":"pith:MZHIGPL7","canonical_record":{"source":{"id":"1010.4522","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-10-21T16:44:15Z","cross_cats_sorted":[],"title_canon_sha256":"9853a426c5f7ff58dcd7dd850e0c1e4304cd009566710632243c7ffe44ff578a","abstract_canon_sha256":"e4a2a199d06d3b44fdf243e6fd6df3c29c72622dc2515b8d3a8316602ccd039d"},"schema_version":"1.0"},"canonical_sha256":"664e833d7fb0e87b537cdb5b8be107bad5ff068c6f99d4ed34458e80ca7caac3","source":{"kind":"arxiv","id":"1010.4522","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1010.4522","created_at":"2026-05-18T04:38:58Z"},{"alias_kind":"arxiv_version","alias_value":"1010.4522v1","created_at":"2026-05-18T04:38:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1010.4522","created_at":"2026-05-18T04:38:58Z"},{"alias_kind":"pith_short_12","alias_value":"MZHIGPL7WDUH","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_16","alias_value":"MZHIGPL7WDUHWU34","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_8","alias_value":"MZHIGPL7","created_at":"2026-05-18T12:26:10Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:MZHIGPL7WDUHWU343NNYXYIHXL","target":"record","payload":{"canonical_record":{"source":{"id":"1010.4522","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-10-21T16:44:15Z","cross_cats_sorted":[],"title_canon_sha256":"9853a426c5f7ff58dcd7dd850e0c1e4304cd009566710632243c7ffe44ff578a","abstract_canon_sha256":"e4a2a199d06d3b44fdf243e6fd6df3c29c72622dc2515b8d3a8316602ccd039d"},"schema_version":"1.0"},"canonical_sha256":"664e833d7fb0e87b537cdb5b8be107bad5ff068c6f99d4ed34458e80ca7caac3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:38:58.525847Z","signature_b64":"LwSfsaWvwJ6ZhMEmQCtfT3/jDtLu6YDPtYODsyb2ogFFqaj53Lk6X2tHIZxlxl2x+xudkrIMxwogdDx61A8EBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"664e833d7fb0e87b537cdb5b8be107bad5ff068c6f99d4ed34458e80ca7caac3","last_reissued_at":"2026-05-18T04:38:58.525195Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:38:58.525195Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1010.4522","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":"2026-05-18T04:38:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7ylEEqZnq8ZKXuhAQgV+riXsh/k8Hqkjf+jolowX8wYSYZ0vtvTXHZxySHEgIq7jzknJ8Kc3o0uYgaA5+ZN9BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T01:33:06.645395Z"},"content_sha256":"5a6ac96c38ff87d1f9e9c9bb89fc34364a777bb2428ee127c05605b0394bff64","schema_version":"1.0","event_id":"sha256:5a6ac96c38ff87d1f9e9c9bb89fc34364a777bb2428ee127c05605b0394bff64"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:MZHIGPL7WDUHWU343NNYXYIHXL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Goldstine-Weston theorem in random normed modules","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Guang Shi","submitted_at":"2010-10-21T16:44:15Z","abstract_excerpt":"This article generalize the classical Goldstine-Weston theorem on normed spaces to one on random normed modules: the image of a random normed module $(E,\\|\\cdot\\|)$ under the random natural embedding $J$ is dense in its double random conjugate space $E^{**}$ with respect to the $(\\epsilon,\\lambda)$ weak star topology; and $J(E)$ is also dense in $E^{**}$ with respect to the locally $L^{0}$-convex weak star topology if $E$ has the countable concatenation property."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.4522","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":"2026-05-18T04:38:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0LwRqGANfkFzNv4fLS4qMkgXjmWsX7DqB/kw8zti8DoeYFG29b5qxiZcxAqFCcyeRVXSS9fXi6i75TZK7PYFAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T01:33:06.645776Z"},"content_sha256":"a20e55957c4e2bec4e12c3535a3751d6fa940b4b6d9fb712be893a6b4fb5c3a0","schema_version":"1.0","event_id":"sha256:a20e55957c4e2bec4e12c3535a3751d6fa940b4b6d9fb712be893a6b4fb5c3a0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MZHIGPL7WDUHWU343NNYXYIHXL/bundle.json","state_url":"https://pith.science/pith/MZHIGPL7WDUHWU343NNYXYIHXL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MZHIGPL7WDUHWU343NNYXYIHXL/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":"2026-05-27T01:33:06Z","links":{"resolver":"https://pith.science/pith/MZHIGPL7WDUHWU343NNYXYIHXL","bundle":"https://pith.science/pith/MZHIGPL7WDUHWU343NNYXYIHXL/bundle.json","state":"https://pith.science/pith/MZHIGPL7WDUHWU343NNYXYIHXL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MZHIGPL7WDUHWU343NNYXYIHXL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:MZHIGPL7WDUHWU343NNYXYIHXL","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":"e4a2a199d06d3b44fdf243e6fd6df3c29c72622dc2515b8d3a8316602ccd039d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-10-21T16:44:15Z","title_canon_sha256":"9853a426c5f7ff58dcd7dd850e0c1e4304cd009566710632243c7ffe44ff578a"},"schema_version":"1.0","source":{"id":"1010.4522","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1010.4522","created_at":"2026-05-18T04:38:58Z"},{"alias_kind":"arxiv_version","alias_value":"1010.4522v1","created_at":"2026-05-18T04:38:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1010.4522","created_at":"2026-05-18T04:38:58Z"},{"alias_kind":"pith_short_12","alias_value":"MZHIGPL7WDUH","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_16","alias_value":"MZHIGPL7WDUHWU34","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_8","alias_value":"MZHIGPL7","created_at":"2026-05-18T12:26:10Z"}],"graph_snapshots":[{"event_id":"sha256:a20e55957c4e2bec4e12c3535a3751d6fa940b4b6d9fb712be893a6b4fb5c3a0","target":"graph","created_at":"2026-05-18T04:38:58Z","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":"This article generalize the classical Goldstine-Weston theorem on normed spaces to one on random normed modules: the image of a random normed module $(E,\\|\\cdot\\|)$ under the random natural embedding $J$ is dense in its double random conjugate space $E^{**}$ with respect to the $(\\epsilon,\\lambda)$ weak star topology; and $J(E)$ is also dense in $E^{**}$ with respect to the locally $L^{0}$-convex weak star topology if $E$ has the countable concatenation property.","authors_text":"Guang Shi","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-10-21T16:44:15Z","title":"The Goldstine-Weston theorem in random normed modules"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.4522","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":"sha256:5a6ac96c38ff87d1f9e9c9bb89fc34364a777bb2428ee127c05605b0394bff64","target":"record","created_at":"2026-05-18T04:38:58Z","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":"e4a2a199d06d3b44fdf243e6fd6df3c29c72622dc2515b8d3a8316602ccd039d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-10-21T16:44:15Z","title_canon_sha256":"9853a426c5f7ff58dcd7dd850e0c1e4304cd009566710632243c7ffe44ff578a"},"schema_version":"1.0","source":{"id":"1010.4522","kind":"arxiv","version":1}},"canonical_sha256":"664e833d7fb0e87b537cdb5b8be107bad5ff068c6f99d4ed34458e80ca7caac3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"664e833d7fb0e87b537cdb5b8be107bad5ff068c6f99d4ed34458e80ca7caac3","first_computed_at":"2026-05-18T04:38:58.525195Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:38:58.525195Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LwSfsaWvwJ6ZhMEmQCtfT3/jDtLu6YDPtYODsyb2ogFFqaj53Lk6X2tHIZxlxl2x+xudkrIMxwogdDx61A8EBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:38:58.525847Z","signed_message":"canonical_sha256_bytes"},"source_id":"1010.4522","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5a6ac96c38ff87d1f9e9c9bb89fc34364a777bb2428ee127c05605b0394bff64","sha256:a20e55957c4e2bec4e12c3535a3751d6fa940b4b6d9fb712be893a6b4fb5c3a0"],"state_sha256":"b0e1a4e3994faef5d2bd404f9548649de3df919790a3d9edbd92db1460f638e0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VqfITIIRwf1lZPqlQ5zCZH58Wjc1exQFRXf10PoIIvZPAvnbmyp479RMRtrEBVhhuaIDwGCMbY2MwOK1oWH2AA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T01:33:06.648370Z","bundle_sha256":"7000c14f47d334563e71addd63bfa8541cc453b490a65cb350c030f14ce8389d"}}