um-Topology in Multi-Normed Vector Lattices

Abstract

Let M = {mλ}λ∈ \Lambda 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 m −→ x if m_λ(xα − x) → 0 for all λ ∈ \Lambda. A net xα in an MNVL X = (X,M) is said to be unbounded m-convergent (or um-convergent) to x if |xα − x| ∧ u m −→ 0 for all u ∈ X+. um-Convergence generalizes un-convergence (Deng et al. in Positivity 21:963–974, 2017; Kandi´c et al. in J Math Anal Appl 451:259–279, 2017) and uaw-convergence (Zabeti in Positivity, 2017.doi:10.1007/s11117-017-0524-7), and specializes up-convergence (Aydın et al. in Unbounded p-convergence in lattice-normed vector lattices. arXiv:1609.05301) and uτ -convergence (Dabboorasad et al. in uτ-Convergence in locally solid vector lattices. arXiv:1706.02006v3). 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 every m-bounded and um-closed set is um-compact iff the space is atomic and has Lebesgue’s and Levi’s properties.