Statistical Convergence of Double Sequences in Locally Solid Riesz Spaces

S A Mohiuddine,M Mursaleen,Abdullah Alotaibi

doi:10.1155/2012/719729

Abstract

Recently, the notion of statistical convergence is studied in a locally solid Riesz space by Albayrak and Pehlivan (2012). In this paper, we define and study statisticalτ-convergence, statisticalτ-Cauchy andS∗(τ)-convergence of double sequences in a locally solid Riesz space.

Highlights

Introduction and PreliminariesThe notion of statistical convergence was introduced by Fast 1 and Steinhaus 2 independently in the same year 1951
We say that a double sequence x xjk in X is statistically τ-bounded if for every τ-neighborhood U of zero there exists some λ > 0, such that the set j, k, j ≤ m, k ≤ n : λxjk ∈/ U
In 30, it was shown that a real double sequence x xjk is statistically convergent to a number if and only if there exists a subset K { j, k } ⊆ N × N, j, k 1, 2, . . . such that δ2 K 1 and limj,k → ∞ xjk

Summary

Introduction

Introduction and PreliminariesThe notion of statistical convergence was introduced by Fast 1 and Steinhaus 2 independently in the same year 1951. A real double sequence x xjk is said to be statistically convergent see 28–30 to the number if for each > 0, the set j, k , j ≤ m, k ≤ n : xjk − ≥

Results

Conclusion