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

Abstract

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