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Completeness of a Normed Space via Strong p-Cesàro Summability

Fernando León-Saavedra, Soledad Moreno-Pulido, Antonio Sala-Pérez

Abstract

In this paper we will characterize the completeness and barrelledness of a normed space through the strong p-Cesàro summability of series. A new characterization of weakly unconditionally Cauchy series and unconditionally convergent series through the strong p-Cesàro summability is obtained.

Completeness of a Normed Space via Strong p-Cesàro Summability

Abstract

In this paper we will characterize the completeness and barrelledness of a normed space through the strong p-Cesàro summability of series. A new characterization of weakly unconditionally Cauchy series and unconditionally convergent series through the strong p-Cesàro summability is obtained.
Paper Structure (5 sections, 10 theorems, 39 equations)

This paper contains 5 sections, 10 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Some preliminary results
  3. The strong $p-$Cesàro summability space
  4. The space of weak $\textrm{w}_p-$ summability
  5. The weak$^{\ast}$ $\textrm{w}_p-$ summability space

Key Result

Proposition 2.2

Let $0<p<\infty$ and $(x_k)_k$ be a sequence in a normed space $X$ such that for some increasing subsequence $(n_j)\subset \mathbb{N}$, $\lim_j\frac{1}{n_j}\sum_{k=1}^{n_j}\|x_k\|^p=+\infty$. Then, $(x_k)_k$ is not strongly $p-$Cesàro summable to any $L\in X$.

Theorems & Definitions (16)

  • Example 2.1
  • Proposition 2.2
  • Proposition 2.3: Connor Connor88
  • Example 2.4
  • Proposition 2.5
  • Remark 2.6
  • Example 2.7
  • Proposition 2.8
  • Theorem 3.1
  • Remark 3.2
  • ...and 6 more