Unbounded Order Convergence and Universal Completions
We characterize vector lattices in which unbounded order convergence is eventually order bounded. Among other things, the characterization provides a solution to \cite[Probl.23]{Az}.
math.FA↗
arXiv subjects
E. Y. Emelyanov
Publications and source records attributed to E. Y. Emelyanov.
Komlos Properties in Banach Lattices
Several Komlós like properties in Banach lattices are investigated. We prove that $C(K)$ fails the $oo$-pre-Komlós property, assuming that the compact Hausdorff space $K$ has a nonempty separable open subset $U$ without isolated points such that every $u\in U$ has countable neighborhood base. We prove also that for any infinite dimensional Banach lattice $E$ there is an unbounded convex $uo$-pre-Komlós set $C\subseteq E_+$ which is not $uo$-Komlós.
math.FA↗
$um$-Topology in multi-normed vector lattices
Let $\mathcal{M}=\{m_λ\}_{λ\inΛ}$ 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 $x_α\xrightarrow{\mathrm{m}} x$ if $m_λ(x_α-x)\to 0$ for all $λ\inΛ$. A net $x_α$ in an MNVL $X=(X,\mathcal{M})$ is said to be unbounded $m$-convergent (or $um$-convergent) to $x$ if $\lvert x_α-x \rvert\wedge u \xrightarrow{\mathrm{m}} 0$ for all $u\in X_+$. $um$-Convergence generalizes $un$-convergence \cite{DOT,KMT} and $uaw$-convergence \cite{Zab}, and specializes $up$-convergence \cite{AEEM1} and $uτ$-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 $σ$-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.
math.FA↗
Order convergence in infinite-dimensional vector lattices is not topological
In this note, we show that the order convergence in a vector lattice $X$ is not topological unless $\dim X<\infty$. Furthermore, we show that, in atomic order continuous Banach lattices, the order convergence is topological on order intervals.
math.FA↗