$um$-Topology in multi-normed vector lattices

Abstract

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 $x_{\alpha }\stackrel{\mathrm{m}}{\to }x$ 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 $|x_{\alpha }-x|\wedge u\stackrel{\mathrm{m}}{\to }0$ for all $u\in X_{+}$. $um$-Convergence generalizes $un$-convergence \cite{DOT,KMT} and $uaw$-convergence \cite{Zab}, and specializes $up$-convergence \cite{AEEM1} and $u\tau$-convergence \cite{DEM2}. $um$-Convergence is always topological, whose corresponding topology is called unbounded $m$-topology (or $um$-topology). We show that, for an $m$-complete metrizable MNVL $(X,\mathcal{M})$, 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 $\sigma$-Lebesgue and $\sigma$-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.

References in corpus (3)

• Order convergence in infinite-dimensional vector lattices is not topological
• Duality for unbounded order convergence and applications
• Compact-Like Operators in Lattice-Normed Spaces