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It may be viewed as a generalization of almost everywhere convergence to general vector lattices. In this paper, we study a variation of this convergence for Banach lattices. A net $(x_\\alpha)$ in a Banach lattice $X$ is unbounded norm convergent to $x$ if $\\lVert\\lvert x_\\alpha - x\\rvert \\wedge u\\rVert\\to 0$ for all $u\\in X_+$. We show"},"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":"1605.03538","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-05-11T18:18:14Z","cross_cats_sorted":[],"title_canon_sha256":"d74b3d9cc6185be562363fa8007486f162e7e3a2d22c2a092f08ed0d40a64c7e","abstract_canon_sha256":"50f26ed689d374a82285d6337a310bfe53a061fe36c5e16e5b48a32f73ade201"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:15:05.188067Z","signature_b64":"QRLbBCB+6+CPveweePy7Khn4fc2sl5+Vi/n+ohkOn2l8lDqJoYmaRbNC9l1EjkMs7Kjnm9+2+0smzxvJljABBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3cdfa0e8c0dde6b1dd573c1d601f8611ecb88ef496680953feedf4abb11067dc","last_reissued_at":"2026-05-18T01:15:05.187597Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:15:05.187597Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Unbounded Norm Convergence in Banach Lattices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"M. O'Brien, V.G. Troitsky, Y. Deng","submitted_at":"2016-05-11T18:18:14Z","abstract_excerpt":"A net $(x_\\alpha)$ in a vector lattice $X$ is unbounded order convergent to $x \\in X$ if $\\lvert x_\\alpha - x\\rvert \\wedge u$ converges to $0$ in order for all $u\\in X_+$. This convergence has been investigated and applied in several recent papers by Gao et al. It may be viewed as a generalization of almost everywhere convergence to general vector lattices. In this paper, we study a variation of this convergence for Banach lattices. A net $(x_\\alpha)$ in a Banach lattice $X$ is unbounded norm convergent to $x$ if $\\lVert\\lvert x_\\alpha - x\\rvert \\wedge u\\rVert\\to 0$ for all $u\\in X_+$. 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