Statistical Order Convergence and Statistically Relatively Uniform Convergence in Riesz Spaces

Xuemei Xue,Jian Tao

doi:10.1155/2018/9092136

Xuemei Xue, Jian Tao

Open Access

https://doi.org/10.1155/2018/9092136

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Abstract

A new concept of statistically e-uniform Cauchy sequences is introduced to study statistical order convergence, statistically relatively uniform convergence, and norm statistical convergence in Riesz spaces. We prove that, for statistically e-uniform Cauchy sequences, these three kinds of convergence for sequences coincide. Moreover, we show that the statistical order convergence and the statistically relatively uniform convergence need not be equivalent. Finally, we prove that, for monotone sequences in Banach lattices, the norm statistical convergence coincides with the weak statistical convergence.

Highlights

The first idea of statistical convergence goes back to Zygmund’s monograph [1], where Zygmund called it almost convergence
We prove that, for monotone sequences in Banach lattices, the norm statistical convergence coincides with the weak statistical convergence
Sencimen and Pehlivan [15] introduced the concept of statistical order convergence, which is a natural generalization of order convergence

Summary

Introduction

The first idea of statistical convergence goes back to Zygmund’s monograph [1], where Zygmund called it almost convergence. It follows from Theorem 11 that there exists a strictly increasing sequence (kj)j∈N of positive integers such that (xkj )j∈N is an e-uniform Cauchy sequence. Let (xn)n∈N be a statistically e-uniform Cauchy sequence in E.

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Results

Conclusion