The statistically unbounded $\tau$-convergence on locally solid Riesz spaces

Abstract

A sequence $(x_n)$ in a locally solid Riesz space $(E,\tau )$ is said to be statistically unbounded $\tau$-convergent to $x\in E$ if, for every zero neighborhood $U$, $\frac{1}{n}|\{k\le n:|x_k-x|\wedge u\notin U\}|\to 0$ as $n\to \infty$. In this paper, we introduce this concept and give the notions $st$-$u_{\tau }$-closed subset, $st$-$u_{\tau }$-Cauchy sequence, $st$-$u_{\tau }$-continuous and $st$-$u_{\tau }$-complete locally solid vector lattice. Also, we give some relations between the order convergence and the $st$-$u_{\tau }$-convergence.

10