On lacunary double statistical convergence in locally solid Riesz spaces

Ekrem Savas

doi:10.1186/1029-242x-2013-99

Abstract

The concept of statistical convergence is one of the most active areas of research in the field of summability. Most of the new summability methods have relation with this popular method. In this paper, we introduce the concept of double I θ -statistical-τ-convergence which is a more general idea of statistical convergence. We also investigate the ideas of double I θ -statistical-τ-boundedness and double I θ -statistical-τ-Cauchy condition of sequences in the framework of locally solid Riesz space endowed with a topology τ and investigate some of their consequences.MSC:40G15, 40A35, 46A40.

Highlights

The notion of statistical convergence, which is an extension of the idea of usual convergence, was introduced by Fast [ ], Steinhaus [ ] independently in the same year and by Schoenberg [ ]
In [ ] Albayrak and Pehlivan studied this notion in locally solid Riesz spaces
In [ ] Mohiuddine et al introduced the concept of lacunary statistical convergence, lacunary statistically bounded and lacunary statistically Cauchy in the framework of locally solid Riesz spaces

Summary

Introduction

The notion of statistical convergence, which is an extension of the idea of usual convergence, was introduced by Fast [ ], Steinhaus [ ] independently in the same year and by Schoenberg [ ]. We introduce the idea of I-double lacunary statistical convergence in a locally solid Riesz space and study some of its properties by using the mathematical tools of the theory of topological vector spaces. A linear topology τ on a Riesz space L is said to be locally solid if τ has a base at zero consisting of solid sets.

Results

Conclusion