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

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 $p(|x_{\alpha }-x|\wedge u)\stackrel{\tau }{\to }0$ 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.

12 pages. arXiv admin note: text overlap with arXiv:1609.05301, text overlap with arXiv:1706.02006 by other authors