Unbounded $p_τ$-convergence in Lattice-Normed Locally Solids

Abdulla Aydın

Published 2017-10-31Version 1

Let $x_\alpha$ be a net in a lattice-normed locally solid $(X,p,E_\tau)$. 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 lattites under the name $up$-convergence in [AGG,AEEM], for $(X,\lvert \cdot \rvert,X)$ under the name of $uo$-convergence in [GTX], for $(X,\lVert\cdot\rVert,{\mathbb R})$ under the name of $un$-convergence in [DOT,KMT], and also for $(X,p,{\mathbb R}^{X^*})$, where $p(x)[f]:=|f|(|x|)$, under the name $uaw$-convergence in [Z]. In this paper we study general properties of the unbounded $p_\tau$-convergence.