Unbounded $p$-convergence in Lattice-Normed Vector Lattices

TL;DR

This paper explores the properties of unbounded p-convergence in lattice-normed vector lattices, generalizing and unifying various known convergence concepts like uo, un, and uaw convergence.

Contribution

It introduces a general framework for unbounded p-convergence, extending previous specific cases and studying its fundamental properties.

Findings

Unifies various convergence notions under a general framework.

Establishes basic properties and characterizations of unbounded p-convergence.

Provides insights into the structure of lattice-normed vector lattices through this convergence.

Abstract

A net $x_\alpha$ in a lattice-normed vector lattice $(X,p,E)$ is unbounded $p$-convergent to $x\in X$ if $p(|x_\alpha-x|\wedge u)\xrightarrow{o} 0$ for every $u\in X_+$. This convergence has been investigated recently for $(X,p,E)=(X,\lvert\cdot \rvert,X)$ under the name of $uo$-convergence, for $(X,p,E)=(X,\lVert\cdot\rVert,{\mathbb R})$ under the name of $un$-convergence, and also for $(X,p,{\mathbb R}^{X^*})$, where $p(x)[f]:=|f|(|x|)$, under the name $uaw$-convergence. In this paper we study general properties of the unbounded $p$-convergence.