Ideal Convergence and Completeness of a Normed Space

Antonio Sala,María Del Pilar Romero De La Rosa,Fernando León-Saavedra,Francisco Javier Pérez-Fernández

doi:10.3390/math7100897

Abstract

We aim to unify several results which characterize when a series is weakly unconditionally Cauchy (wuc) in terms of the completeness of a convergence space associated to the wuc series. If, additionally, the space is completed for each wuc series, then the underlying space is complete. In the process the existing proofs are simplified and some unanswered questions are solved. This research line was originated in the PhD thesis of the second author. Since then, it has been possible to characterize the completeness of a normed spaces through different convergence subspaces (which are be defined using different kinds of convergence) associated to an unconditionally Cauchy sequence.

Highlights

A sequence in a Banach space X is said to be statistically convergent to a vector L if for any ε > 0 the subset {n : k xn − Lk > ε} has density 0
As a consequence we provide an unified point of view which allows us to solve several unsolved questions
We will show that under reasonable conditions on a given non-trivial ideal, the studied properties do not depend on the ideal that we use to define the convergence spaces associated to the wuc series

Summary

Introduction

A sequence ( xn ) in a Banach space X is said to be statistically convergent to a vector L if for any ε > 0 the subset {n : k xn − Lk > ε} has density 0. The aim of this paper originates in the PhD thesis of the second author [6] who discovered a relationship between properties of a normed space X and some sequence spaces which are called convergence spaces associated to a weakly unconditionally Cauchy series. We will show that under reasonable conditions on a given non-trivial ideal, the studied properties do not depend on the ideal that we use to define the convergence spaces associated to the wuc series This allows us to extend our results for an arbitrary summability method that shares some kind of ideal-convergence on the realm of all bounded sequences. This will allow us to unify the known results and obtain answers to some unresolved questions.

Some Preliminary Results

Main Results

Some Applications