{"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 $u\\in X_+$. We show"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.03538","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"}