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It is shown that the subspace $Y$ is dense in $L^p(E)$. Moreover, the Hahn-Banach theorem's extension $\\bar T_g\\in [L^p(E)]^*$ of the bounded on $Y$ functional $h\\mapsto \\int_E g(x)h(x)\\,dx$, $h\\in Y$, can not be represented in the form $\\bar T_g(h)= \\int_E g(x)h(x)\\,dx$, $h\\in L^p(E)$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1802.08347","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-02-22T23:38:35Z","cross_cats_sorted":[],"title_canon_sha256":"40e533b9421ce83b7842446f828f0b07947430c4a99f866f009256cad5f7a227","abstract_canon_sha256":"8a308d4a1ad70e31d156de5b18db8738d82371901378c2b8cd028b911a245a6c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:56:25.164396Z","signature_b64":"SPRMTms3dfVyka96DzlCUxIuVgbnaCvWVTaZ3HuRxviFxVfnKAZaTnlHg+nYlcluyUQMWr6RzplFLLlq1YXLCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"746f26d1322855ea835b17992a01b5905657d3fd5e3291aed975f96c126e2078","last_reissued_at":"2026-05-17T23:56:25.163991Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:56:25.163991Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the subspace of the $L^p$ space, which is an annihilator of an element not belonging to the dual space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Dmitrii Prokhorov","submitted_at":"2018-02-22T23:38:35Z","abstract_excerpt":"Let $E$ be a Lebesgue measurable subset of ${\\mathbb R}^n$, $p\\in [1,\\infty)$. We consider the subspace $Y\\subset L^p(E)$, which is an annihilator of the Lebesgue measurable ${{\\cal L}^{n}}$-a.e. finite function $g$ that does not belong to the dual space of $L^p(E)$. It is shown that the subspace $Y$ is dense in $L^p(E)$. Moreover, the Hahn-Banach theorem's extension $\\bar T_g\\in [L^p(E)]^*$ of the bounded on $Y$ functional $h\\mapsto \\int_E g(x)h(x)\\,dx$, $h\\in Y$, can not be represented in the form $\\bar T_g(h)= \\int_E g(x)h(x)\\,dx$, $h\\in L^p(E)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.08347","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1802.08347","created_at":"2026-05-17T23:56:25.164054+00:00"},{"alias_kind":"arxiv_version","alias_value":"1802.08347v2","created_at":"2026-05-17T23:56:25.164054+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1802.08347","created_at":"2026-05-17T23:56:25.164054+00:00"},{"alias_kind":"pith_short_12","alias_value":"ORXSNUJSFBK6","created_at":"2026-05-18T12:32:43.782077+00:00"},{"alias_kind":"pith_short_16","alias_value":"ORXSNUJSFBK6VA23","created_at":"2026-05-18T12:32:43.782077+00:00"},{"alias_kind":"pith_short_8","alias_value":"ORXSNUJS","created_at":"2026-05-18T12:32:43.782077+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ORXSNUJSFBK6VA23C6MSUANVSB","json":"https://pith.science/pith/ORXSNUJSFBK6VA23C6MSUANVSB.json","graph_json":"https://pith.science/api/pith-number/ORXSNUJSFBK6VA23C6MSUANVSB/graph.json","events_json":"https://pith.science/api/pith-number/ORXSNUJSFBK6VA23C6MSUANVSB/events.json","paper":"https://pith.science/paper/ORXSNUJS"},"agent_actions":{"view_html":"https://pith.science/pith/ORXSNUJSFBK6VA23C6MSUANVSB","download_json":"https://pith.science/pith/ORXSNUJSFBK6VA23C6MSUANVSB.json","view_paper":"https://pith.science/paper/ORXSNUJS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1802.08347&json=true","fetch_graph":"https://pith.science/api/pith-number/ORXSNUJSFBK6VA23C6MSUANVSB/graph.json","fetch_events":"https://pith.science/api/pith-number/ORXSNUJSFBK6VA23C6MSUANVSB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ORXSNUJSFBK6VA23C6MSUANVSB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ORXSNUJSFBK6VA23C6MSUANVSB/action/storage_attestation","attest_author":"https://pith.science/pith/ORXSNUJSFBK6VA23C6MSUANVSB/action/author_attestation","sign_citation":"https://pith.science/pith/ORXSNUJSFBK6VA23C6MSUANVSB/action/citation_signature","submit_replication":"https://pith.science/pith/ORXSNUJSFBK6VA23C6MSUANVSB/action/replication_record"}},"created_at":"2026-05-17T23:56:25.164054+00:00","updated_at":"2026-05-17T23:56:25.164054+00:00"}