{"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. Y. Emelyanov, M. A. A. Marabeh, Y. A. Dabboorasad","submitted_at":"2017-06-19T00:49:11Z","abstract_excerpt":"Let $\\mathcal{M}=\\{m_\\lambda\\}_{\\lambda\\in\\Lambda}$ be a separating family of lattice seminorms on a vector lattice $X$, then $(X,\\mathcal{M})$ is called a multi-normed vector lattice (or MNVL). 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"},"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"}