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We show that for every sequence $(x_n)_{n\\ge 0}$ in $\\mathcal{X}$ satisfying \\begin{eqnarray*} x_{n+1}-a_nx_n-b_n\\in V\\q(n\\geq 0) \\end{eqnarray*} there exists a unique sequence $(y_n)_{n\\ge 0}$ satisfying the recurrence $y_{n+1}=a_ny_n+b_n\\,\\,(n\\geq 0)$ and for every $q$ "},"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":"1006.1940","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-06-10T02:18:49Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"c1d124927f1895dbb3062fa5b5c8dea926a2cf9a7d8d2a8152a0cc72a3fdf678","abstract_canon_sha256":"b19945f0dccf6e4e79ed601fed0ef3e0012a810bab651264ace49e27b9be840e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:59:38.547640Z","signature_b64":"iM1oyEk4qxROTSx0nPgbWr9GLuDdFDFpSknbYCAEvHIZU2DpIRkPnSNFChYNXPIw8xJF5xPpbhmG9aDkPlM7BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a8b08613d87adda29e333a05dbdc905bd42896409d0e70a588ddd774474f14e0","last_reissued_at":"2026-05-18T03:59:38.546909Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:59:38.546909Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the stability of the first order linear recurrence in topological vector spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.FA","authors_text":"Dorian Popa, Mohammad Sal Moslehian","submitted_at":"2010-06-10T02:18:49Z","abstract_excerpt":"Suppose that $\\mathcal{X}$ is a sequentially complete Hausdorff locally convex space over a scalar field $\\mathbb{K}$, $V$ is a bounded subset of $\\mathcal{X}$, $(a_n)_{n\\ge 0}$ is a sequence in $\\mathbb{K}\\setminus\\{0\\}$ with the property\\ $\\ds\\liminf_{n\\to\\infty} |a_n|>1$ and $(b_n)_{n\\ge 0}$ is a sequence in $\\mathcal{X}$. We show that for every sequence $(x_n)_{n\\ge 0}$ in $\\mathcal{X}$ satisfying \\begin{eqnarray*} x_{n+1}-a_nx_n-b_n\\in V\\q(n\\geq 0) \\end{eqnarray*} there exists a unique sequence $(y_n)_{n\\ge 0}$ satisfying the recurrence $y_{n+1}=a_ny_n+b_n\\,\\,(n\\geq 0)$ and for every $q$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1006.1940","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1006.1940","created_at":"2026-05-18T03:59:38.547020+00:00"},{"alias_kind":"arxiv_version","alias_value":"1006.1940v1","created_at":"2026-05-18T03:59:38.547020+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1006.1940","created_at":"2026-05-18T03:59:38.547020+00:00"},{"alias_kind":"pith_short_12","alias_value":"VCYIME6YPLO2","created_at":"2026-05-18T12:26:15.391820+00:00"},{"alias_kind":"pith_short_16","alias_value":"VCYIME6YPLO2FHRT","created_at":"2026-05-18T12:26:15.391820+00:00"},{"alias_kind":"pith_short_8","alias_value":"VCYIME6Y","created_at":"2026-05-18T12:26:15.391820+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/VCYIME6YPLO2FHRTHIC5XXEQLP","json":"https://pith.science/pith/VCYIME6YPLO2FHRTHIC5XXEQLP.json","graph_json":"https://pith.science/api/pith-number/VCYIME6YPLO2FHRTHIC5XXEQLP/graph.json","events_json":"https://pith.science/api/pith-number/VCYIME6YPLO2FHRTHIC5XXEQLP/events.json","paper":"https://pith.science/paper/VCYIME6Y"},"agent_actions":{"view_html":"https://pith.science/pith/VCYIME6YPLO2FHRTHIC5XXEQLP","download_json":"https://pith.science/pith/VCYIME6YPLO2FHRTHIC5XXEQLP.json","view_paper":"https://pith.science/paper/VCYIME6Y","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1006.1940&json=true","fetch_graph":"https://pith.science/api/pith-number/VCYIME6YPLO2FHRTHIC5XXEQLP/graph.json","fetch_events":"https://pith.science/api/pith-number/VCYIME6YPLO2FHRTHIC5XXEQLP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VCYIME6YPLO2FHRTHIC5XXEQLP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VCYIME6YPLO2FHRTHIC5XXEQLP/action/storage_attestation","attest_author":"https://pith.science/pith/VCYIME6YPLO2FHRTHIC5XXEQLP/action/author_attestation","sign_citation":"https://pith.science/pith/VCYIME6YPLO2FHRTHIC5XXEQLP/action/citation_signature","submit_replication":"https://pith.science/pith/VCYIME6YPLO2FHRTHIC5XXEQLP/action/replication_record"}},"created_at":"2026-05-18T03:59:38.547020+00:00","updated_at":"2026-05-18T03:59:38.547020+00:00"}