{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:R3SACWIOVPLCXREJHBNKRY2FCL","short_pith_number":"pith:R3SACWIO","canonical_record":{"source":{"id":"2109.13512","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2021-09-28T06:26:25Z","cross_cats_sorted":[],"title_canon_sha256":"d1165cf7307735bb50ac729d1f6cce5f378f359734d48295f34e934f8f5520a4","abstract_canon_sha256":"24a97c365e585ef4f65eb8a928868450d260d8f5998d3ead0bb08e076a38e67c"},"schema_version":"1.0"},"canonical_sha256":"8ee401590eabd62bc489385aa8e34512c079779d7c42b1262cef65d44808ac68","source":{"kind":"arxiv","id":"2109.13512","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2109.13512","created_at":"2026-07-05T04:23:37Z"},{"alias_kind":"arxiv_version","alias_value":"2109.13512v4","created_at":"2026-07-05T04:23:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2109.13512","created_at":"2026-07-05T04:23:37Z"},{"alias_kind":"pith_short_12","alias_value":"R3SACWIOVPLC","created_at":"2026-07-05T04:23:37Z"},{"alias_kind":"pith_short_16","alias_value":"R3SACWIOVPLCXREJ","created_at":"2026-07-05T04:23:37Z"},{"alias_kind":"pith_short_8","alias_value":"R3SACWIO","created_at":"2026-07-05T04:23:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:R3SACWIOVPLCXREJHBNKRY2FCL","target":"record","payload":{"canonical_record":{"source":{"id":"2109.13512","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2021-09-28T06:26:25Z","cross_cats_sorted":[],"title_canon_sha256":"d1165cf7307735bb50ac729d1f6cce5f378f359734d48295f34e934f8f5520a4","abstract_canon_sha256":"24a97c365e585ef4f65eb8a928868450d260d8f5998d3ead0bb08e076a38e67c"},"schema_version":"1.0"},"canonical_sha256":"8ee401590eabd62bc489385aa8e34512c079779d7c42b1262cef65d44808ac68","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:23:37.883547Z","signature_b64":"UmRNO038jqPTH+EZ+GHCR6cTrVA9kM+LAKuEJZDAzxA42BXwfD/jFrDDQAASsO1sp98UXnlLW4XIAOVmUxjZBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8ee401590eabd62bc489385aa8e34512c079779d7c42b1262cef65d44808ac68","last_reissued_at":"2026-07-05T04:23:37.883140Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:23:37.883140Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2109.13512","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T04:23:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"A31oAt7UCuVdg7+hiMu1w9o49CY14o3jrV1aJAdZl12At8VlkXUnyVLUdYJ15zKK4Qtwt/nnnQZnyF6ZqsTWDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-05T13:30:40.429226Z"},"content_sha256":"b77cace7a3770660e134566a83515235aae2a41a5f3b4783d0bce3191dfdf846","schema_version":"1.0","event_id":"sha256:b77cace7a3770660e134566a83515235aae2a41a5f3b4783d0bce3191dfdf846"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:R3SACWIOVPLCXREJHBNKRY2FCL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Neural Networks in Fr\\'echet spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Fred Espen Benth, Luca Galimberti, Nils Detering","submitted_at":"2021-09-28T06:26:25Z","abstract_excerpt":"We define a neural network in infinite dimensional spaces for which we can show the universal approximation property. Indeed, we derive approximation results for continuous functions from a Fr\\'echet space $\\X$ into a Banach space $\\Y$. The approximation results are generalising the well known universal approximation theorem for continuous functions from $\\mathbb{R}^n$ to $\\mathbb{R}$, where approximation is done with (multilayer) neural networks [15, 25, 18, 29]. Our infinite dimensional networks are constructed using activation functions being nonlinear operators and affine transforms. Sever"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2109.13512","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2109.13512/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T04:23:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0TNRQwmjHlUF7KZfbETQlNff+nAuom0eQ2Sy50Bg4og16AYRrlIKEEHNr+KaZ9lAQJDZAxh9zhKZl2DleuJWCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-05T13:30:40.429650Z"},"content_sha256":"c1cf01b54a994af88fe13af0f35bcf9ddd46bf8d3b745a5ebb69a629e9de7822","schema_version":"1.0","event_id":"sha256:c1cf01b54a994af88fe13af0f35bcf9ddd46bf8d3b745a5ebb69a629e9de7822"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/R3SACWIOVPLCXREJHBNKRY2FCL/bundle.json","state_url":"https://pith.science/pith/R3SACWIOVPLCXREJHBNKRY2FCL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/R3SACWIOVPLCXREJHBNKRY2FCL/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":"2026-07-05T13:30:40Z","links":{"resolver":"https://pith.science/pith/R3SACWIOVPLCXREJHBNKRY2FCL","bundle":"https://pith.science/pith/R3SACWIOVPLCXREJHBNKRY2FCL/bundle.json","state":"https://pith.science/pith/R3SACWIOVPLCXREJHBNKRY2FCL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/R3SACWIOVPLCXREJHBNKRY2FCL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:R3SACWIOVPLCXREJHBNKRY2FCL","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":"24a97c365e585ef4f65eb8a928868450d260d8f5998d3ead0bb08e076a38e67c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2021-09-28T06:26:25Z","title_canon_sha256":"d1165cf7307735bb50ac729d1f6cce5f378f359734d48295f34e934f8f5520a4"},"schema_version":"1.0","source":{"id":"2109.13512","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2109.13512","created_at":"2026-07-05T04:23:37Z"},{"alias_kind":"arxiv_version","alias_value":"2109.13512v4","created_at":"2026-07-05T04:23:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2109.13512","created_at":"2026-07-05T04:23:37Z"},{"alias_kind":"pith_short_12","alias_value":"R3SACWIOVPLC","created_at":"2026-07-05T04:23:37Z"},{"alias_kind":"pith_short_16","alias_value":"R3SACWIOVPLCXREJ","created_at":"2026-07-05T04:23:37Z"},{"alias_kind":"pith_short_8","alias_value":"R3SACWIO","created_at":"2026-07-05T04:23:37Z"}],"graph_snapshots":[{"event_id":"sha256:c1cf01b54a994af88fe13af0f35bcf9ddd46bf8d3b745a5ebb69a629e9de7822","target":"graph","created_at":"2026-07-05T04:23:37Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2109.13512/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We define a neural network in infinite dimensional spaces for which we can show the universal approximation property. Indeed, we derive approximation results for continuous functions from a Fr\\'echet space $\\X$ into a Banach space $\\Y$. The approximation results are generalising the well known universal approximation theorem for continuous functions from $\\mathbb{R}^n$ to $\\mathbb{R}$, where approximation is done with (multilayer) neural networks [15, 25, 18, 29]. Our infinite dimensional networks are constructed using activation functions being nonlinear operators and affine transforms. Sever","authors_text":"Fred Espen Benth, Luca Galimberti, Nils Detering","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2021-09-28T06:26:25Z","title":"Neural Networks in Fr\\'echet spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2109.13512","kind":"arxiv","version":4},"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:b77cace7a3770660e134566a83515235aae2a41a5f3b4783d0bce3191dfdf846","target":"record","created_at":"2026-07-05T04:23:37Z","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":"24a97c365e585ef4f65eb8a928868450d260d8f5998d3ead0bb08e076a38e67c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2021-09-28T06:26:25Z","title_canon_sha256":"d1165cf7307735bb50ac729d1f6cce5f378f359734d48295f34e934f8f5520a4"},"schema_version":"1.0","source":{"id":"2109.13512","kind":"arxiv","version":4}},"canonical_sha256":"8ee401590eabd62bc489385aa8e34512c079779d7c42b1262cef65d44808ac68","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8ee401590eabd62bc489385aa8e34512c079779d7c42b1262cef65d44808ac68","first_computed_at":"2026-07-05T04:23:37.883140Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T04:23:37.883140Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"UmRNO038jqPTH+EZ+GHCR6cTrVA9kM+LAKuEJZDAzxA42BXwfD/jFrDDQAASsO1sp98UXnlLW4XIAOVmUxjZBg==","signature_status":"signed_v1","signed_at":"2026-07-05T04:23:37.883547Z","signed_message":"canonical_sha256_bytes"},"source_id":"2109.13512","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b77cace7a3770660e134566a83515235aae2a41a5f3b4783d0bce3191dfdf846","sha256:c1cf01b54a994af88fe13af0f35bcf9ddd46bf8d3b745a5ebb69a629e9de7822"],"state_sha256":"662c51184bbab993fd534dfa5c10e8014160842e2ca130b02dacf247faf5b25f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vkzOsngoAfPlrwN8E0wvvMO/o42Vc52HthRfEjDW7OrkyYR1zBzg7EwOhlpS3woKL8oWkSNzeWm1n/50rQODCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-05T13:30:40.432739Z","bundle_sha256":"39c935d289af29486072b5c9e9ddd53becc82a4b87a6fb4bbb1d968d8c00dfda"}}