2017/06/19 by Y. A. Dabboorasad, Dabboorasad, Y. A., Eduard Emelyanov +3
Computer Science · Mathematics · #Advanced Banach Space Theory #Approximation Theory and Sequence Spaces #FOS: Mathematics #Functional Analysis (math.FA) #Optimization and Variational Analysis
paper · pdf · doi:10.48550/arxiv.1706.05755
openalex publication_date 2017/06/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let M=\mλ\λ∈Λ be a separating family of lattice seminorms on a vector lattice X, then (X,M) is called a multi-normed vector lattice (or MNVL). We write xα\xrightarrowm x if mλ(xα-x)→ 0 for all λ∈Λ. A net xα in an MNVL X=(X,M) is said to be unbounded m-convergent (or um-convergent) to x if | xα-x |\wedge u \xrightarrowm 0 for all u∈ X+. um-Convergence generalizes un-convergence \citeDOT,KMT and uaw-convergence \citeZab, and specializes up-convergence \citeAEEM1 and uτ-convergence \citeDEM2. um-Convergence is always topological, whose corresponding topology is called unbounded m-topology (or um-topology). We show that, for an m-complete metrizable MNVL (X,M), the um-topology is metrizable iff X has a countable topological orthogonal system. In terms of um-completeness, we present a characterization of MNVLs possessing both Lebesgue's and Levi's properties. Then, we characterize MNVLs possessing simultaneously the σ-Lebesgue and σ-Levi properties in terms of sequential um-completeness. Finally, we prove that any m-bounded and um-closed set is um-compact iff the space is atomic and has Lebesgue's and Levi's properties.