Related papers: Uniform Kadec-Klee Property in Banach lattices

We show that if a rearrangement invariant Banach function space $E$ on the positive semi-axis satisfies a non-trivial lower $q-$ estimate with constant $1$ then the corresponding space $E(\nm)$ of $\tau-$measurable operators, affiliated…

A net $(x_\alpha)$ in a Banach lattice $X$ is said to un-converge to a vector $x$ if $\bigl\lVert\lvert x_\alpha-x\rvert\wedge u\bigr\rVert\to 0$ for every $u\in X_+$. In this paper, we investigate un-topology, i.e., the topology that…

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…

A well-known theorem due to R. C. James states that a Banach space is reflexive if and only if every bounded linear functional attains its norm. In this note we study Banach lattices on which every (real-valued) lattice homomorphism attains…

A net $(x_\gamma)_{\gamma\in\Gamma}$ in a locally solid Riesz space $(X,\tau)$ is said to be unbounded $\tau$-convergent to $x$ if $|x_\gamma-x|\wedge u\mathop{\overset{\tau}{\longrightarrow}} 0$ for all $u\in X_+$. We recall that there is…

Given a map $f \colon E \longrightarrow F$ between Banach spaces (or Banach lattices), a set $A$ of $E$-valued bounded sequences, ${\bf x} \in A$ and a vector topology $\tau$ on $F$, we investigate the existence of an infinite dimensional…

Let $\mathcal{X} = \{ X_{\gamma} \}_{\gamma \in \Gamma}$ be a family of Banach spaces and let $\mathcal{E}$ be a Banach sequence space defined on $\Gamma$. The main aim of this work is to investigate the abstract Kadets--Klee properties,…

In this paper, we generalize the concept of unbounded norm (un) convergence: let $X$ be a normed lattice and $Y$ a vector lattice such that $X$ is an order dense ideal in $Y$; we say that a net $(y_\alpha)$ un-converges to $y$ in $Y$ with…

Let $x_\alpha$ be a net in a locally solid vector lattice $(X,\tau)$; we say that $x_\alpha$ is unbounded $\tau$-convergent to a vector $x\in X$ if $\lvert x_\alpha-x \rvert\wedge w \xrightarrow{\tau} 0$ for all $w\in X_+$. In this paper,…

A net $(x_\alpha)$ in a vector lattice $X$ is unbounded order convergent to $x \in X$ if $\lvert x_\alpha - x\rvert \wedge u$ converges to $0$ in order for all $u\in X_+$. This convergence has been investigated and applied in several recent…

It is well known that every (real or complex) normed linear space $L$ is isometrically embeddable into $C(X)$ for some compact Hausdorff space $X$. Here $X$ is the closed unit ball of $L^*$ (the set of all continuous scalar-valued linear…

In this paper, we continue the investigation of topological properties of unbounded norm (un-)topology in normed lattices. We characterize separability and second countability of un-topology in terms of properties of the underlying normed…

In this paper we study the structure of the set $\mbox{Hom}(X,\mathbb{R})$ of all lattice homomorphisms from a Banach lattice $X$ into $\mathbb{R}$. Using the relation among lattice homomorphisms and disjoint families, we prove that the…

If $X$ is a closed subspace of a Banach space $L$ which embeds into a Banach lattice not containing $\ell_\infty^n$'s uniformly and $L/X$ contains $\ell_\infty^n$'s uniformly, then $X$ cannot have local unconditional structure in the sense…

An important consequence of the Hahn-Banach Theorem says that on any locally convex Hausdorff topological space $X$, there are sufficiently many continuous linear functionals to separate points of $X$. In the paper, we establish a `local'…

Several Koml\'os like properties in Banach lattices are investigated. We prove that $C(K)$ fails the $oo$-pre-Koml\'os property, assuming that the compact Hausdorff space $K$ has a nonempty separable open subset $U$ without isolated points…

A topology on a nonempty set $X$ specifies a natural subset of $\mathcal{P}(X)$. By identifying $\mathcal{P}(\mathcal{P}(X))$ with the totally disconnected compact Hausdorff space $2^{\mathcal{P}(X)}$, the lattice $Top(X)$ of all topologies…

As a generalization of almost everywhere convergence to vector lattices, unbounded order convergence has garnered much attention. The concept of boundedly uo-complete Banach lattices was introduced by N. Gao and F. Xanthos, and has been…

A net $(x_\alpha)$ in a vector lattice $X$ is said to be {unbounded order convergent} (or uo-convergent, for short) to $x\in X$ if the net $(\abs{x_\alpha-x}\wedge y)$ converges to 0 in order for all $y\in X_+$. In this paper, we study…

We are interested in the question when a Banach space $X$ with an unconditional basis is isomorphic (as a Banach space) to an order-continuous nonatomic Banach lattice. We show that this is the case if and only if $X$ is isomorphic as a…