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

A. Aydın, E. Yu. Emelyanov, N. Erkurşun Özcan, M. A. A. Marabeh

Published 2016-09-17Version 1

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 in \cite{GTX}, for $(X,p,E)=(X,\|\cdot\|,{\mathbb R})$ under the name of $un$-convergence in \cite{DOT,KMT}, and also for $(X,p,{\mathbb R}^{X^*})$, where $p(x)[f]:=|f|(|x|)$ under the name $uaw$-convergence in \cite{Z}. In this paper we study general properties of the unbounded $p$-convergence and of mixed-normed spaces.