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We write $x_\\alpha \\xrightarrow{\\mathrm{m}} x$ if $m_\\lambda(x_\\alpha-x)\\to 0$ for all $\\lambda\\in\\Lambda$. A net $x_\\alpha$ in an MNVL $X=(X,\\mathcal{M})$ is said to be unbounded $m$-convergent (or $um$-convergent) to $x$ if $\\lvert x_\\alpha-x \\rvert\\wedge u \\xrightarrow{\\mathrm{m}} 0$ for all $u\\in X_+$. $um$-Convergence generalizes $un$-convergence \\cite{DOT,KMT} and $uaw$-convergence \\cite{Zab}, an"},"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":"1706.05755","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2017-06-19T00:49:11Z","cross_cats_sorted":[],"title_canon_sha256":"78c030ddb67c2945f2ae6e2111bf28524b881d1566b54bfe611b676a2fa80bac","abstract_canon_sha256":"6b6e2b7cfc9d65f6d8b3e88b535309db0f476ba8968a10fd8f7ca20276dfb560"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:42:04.272685Z","signature_b64":"R5dCzZWpzcMBH1OtVfskAJUH0n7nuUqwhRVu7SWr85Wf7El+Fsvu7CfDuJipk7fFvcMRKDJT4tsg37D4ALB0DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3f2df3d915001cdbb586f09296372b8ac4bcb838b966c415d51ff53b88d37258","last_reissued_at":"2026-05-18T00:42:04.272001Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:42:04.272001Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"$um$-Topology in multi-normed vector lattices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"E. 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A net $x_\\alpha$ in an MNVL $X=(X,\\mathcal{M})$ is said to be unbounded $m$-convergent (or $um$-convergent) to $x$ if $\\lvert x_\\alpha-x \\rvert\\wedge u \\xrightarrow{\\mathrm{m}} 0$ for all $u\\in X_+$. $um$-Convergence generalizes $un$-convergence \\cite{DOT,KMT} and $uaw$-convergence \\cite{Zab}, an"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.05755","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":"1706.05755","created_at":"2026-05-18T00:42:04.272126+00:00"},{"alias_kind":"arxiv_version","alias_value":"1706.05755v2","created_at":"2026-05-18T00:42:04.272126+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.05755","created_at":"2026-05-18T00:42:04.272126+00:00"},{"alias_kind":"pith_short_12","alias_value":"H4W7HWIVAAON","created_at":"2026-05-18T12:31:18.294218+00:00"},{"alias_kind":"pith_short_16","alias_value":"H4W7HWIVAAONXNMG","created_at":"2026-05-18T12:31:18.294218+00:00"},{"alias_kind":"pith_short_8","alias_value":"H4W7HWIV","created_at":"2026-05-18T12:31:18.294218+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/H4W7HWIVAAONXNMG6CJJMNZLRL","json":"https://pith.science/pith/H4W7HWIVAAONXNMG6CJJMNZLRL.json","graph_json":"https://pith.science/api/pith-number/H4W7HWIVAAONXNMG6CJJMNZLRL/graph.json","events_json":"https://pith.science/api/pith-number/H4W7HWIVAAONXNMG6CJJMNZLRL/events.json","paper":"https://pith.science/paper/H4W7HWIV"},"agent_actions":{"view_html":"https://pith.science/pith/H4W7HWIVAAONXNMG6CJJMNZLRL","download_json":"https://pith.science/pith/H4W7HWIVAAONXNMG6CJJMNZLRL.json","view_paper":"https://pith.science/paper/H4W7HWIV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1706.05755&json=true","fetch_graph":"https://pith.science/api/pith-number/H4W7HWIVAAONXNMG6CJJMNZLRL/graph.json","fetch_events":"https://pith.science/api/pith-number/H4W7HWIVAAONXNMG6CJJMNZLRL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/H4W7HWIVAAONXNMG6CJJMNZLRL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/H4W7HWIVAAONXNMG6CJJMNZLRL/action/storage_attestation","attest_author":"https://pith.science/pith/H4W7HWIVAAONXNMG6CJJMNZLRL/action/author_attestation","sign_citation":"https://pith.science/pith/H4W7HWIVAAONXNMG6CJJMNZLRL/action/citation_signature","submit_replication":"https://pith.science/pith/H4W7HWIVAAONXNMG6CJJMNZLRL/action/replication_record"}},"created_at":"2026-05-18T00:42:04.272126+00:00","updated_at":"2026-05-18T00:42:04.272126+00:00"}