Unbounded $p_\tau$-Convergence in Vector Lattices Normed by Locally Solid Lattices

TL;DR

This paper introduces and analyzes the properties of unbounded $p_ au$-convergence in vector lattices normed by locally solid lattices, unifying various types of convergence studied recently.

Contribution

It generalizes and studies the fundamental properties of unbounded $p_ au$-convergence, encompassing recent notions like $up$, $uo$, and $un$-convergence.

Findings

Established basic properties of unbounded $p_ au$-convergence.

Unified various recent convergence concepts under a general framework.

Provided conditions for convergence and related topological properties.

Abstract

Let $(x_\alpha)$ be a net in a vector lattice normed by locally solid lattice $(X,p,E_\tau)$. We say that $(x_\alpha)$ is unbounded $p_\tau$-convergent to $x\in X$ if $p(\lvert x_\alpha-x\rvert\wedge u)\xrightarrow{\tau} 0$ for every $u\in X_+$. This convergence has been studied recently for lattice-normed vector lattices as the $up$-convergence in \cite{AGG,AEEM,AEEM2}, the $uo$-convergence in \cite{GTX}, and, as the $un$-convergence in \cite{DOT,GX,GTX,KMT,Tr2}. In this paper, we study the general properties of the unbounded $p_\tau$-convergence.