Invariant Summability and Unconditionally Cauchy Series

Nimet Pancaroğlu Akın

doi:10.28924/2291-8639-18-2020-663

Abstract

In this study, we will give new characterizations of weakly unconditionally Cauchy series and unconditionally convergent series through summability obtained by the invariant convergence.

Highlights

Let σ be a mapping of the positive integers into itself
The above sets endowed with the sup norm and they will be called the space of convergence and the space of weak convergence associated to the series i xi
A series i xi in a normed space X is said to be a weakly unconditionally Cauchy(wuc) if for each ε > 0 and f ∈ X∗, an n0 ∈ N can be found such that for each finite subset F ⊂ N with F ∩ {1, . . . , n0} = ∅ is i∈F |f (xi)| < ε

Summary

Introduction

Let σ be a mapping of the positive integers into itself. A continuous linear functional φ on m, the space of real bounded sequences, is said to be an invariant mean or a σ mean, if and only if, (1) φ(x) ≥ 0, when the sequence x = (xj) is such that xj ≥ 0 for all j, (2) φ(e) = 1,where e = (1, 1, 1....), (3) φ(xσ(j)) = φ(x) for all x ∈ m. Φ extends the limit functional on c, the space of convergent sequences, in the sense that φ(x) = lim x for all x ∈ c. In case σ is translation mappings σ(j) = j + 1, the σ mean is often called a Banach limit and Vσ, the set of bounded sequences all of whose invariant means are equal, is the set of almost convergent sequences. Key words and phrases. unconditionally Cauchy series; invariant convergence; invariant convergent series

It can be shown that

Main Results

We have that there

Now we consider the vector space