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We prove that every norm bounded uo-convergent net in $X^*$ is $w^*$-convergent iff $X$ has order continuous norm, and that every $w^*$-convergent net in $X^*$ is uo-convergent iff $X$ is atomic with order continuous norm. We also characterize among $\\sigma$-order complet"},"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":"1310.4438","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-10-16T16:28:57Z","cross_cats_sorted":[],"title_canon_sha256":"ded18ba7f1f1ac1928db232327e130d9f2db701346d0ea52be495d4dfd79513d","abstract_canon_sha256":"8281751c58d48f73861675a5b38030e8d9d88f95012db5407b11752fecb4f109"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:46:02.948583Z","signature_b64":"kRXUb7TqKIO1EjPLWE2OTJZLT3AiCPAgeNVuFIyfwDLKW2c9i6rMbQ4AgWcDxUVaXy0GhxF+1kb8iU/+HzmEDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a6a734e20197ebc377975005096703a8c41de253ec6cb67e88ac40708f4ec265","last_reissued_at":"2026-05-18T00:46:02.948019Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:46:02.948019Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Unbounded order convergence in dual spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Niushan Gao","submitted_at":"2013-10-16T16:28:57Z","abstract_excerpt":"A net $(x_\\alpha)$ in a vector lattice $X$ is said to be {unbounded order convergent} (or uo-convergent, for short) to $x\\in X$ if the net $(\\abs{x_\\alpha-x}\\wedge y)$ converges to 0 in order for all $y\\in X_+$. In this paper, we study unbounded order convergence in dual spaces of Banach lattices. Let $X$ be a Banach lattice. We prove that every norm bounded uo-convergent net in $X^*$ is $w^*$-convergent iff $X$ has order continuous norm, and that every $w^*$-convergent net in $X^*$ is uo-convergent iff $X$ is atomic with order continuous norm. 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