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Characterizations of a Banach Space through the Strong Lacunary and the Lacunary Statistical Summabilities

Soledad Moreno-Pulido, Giuseppina Barbieri, Fernando León-Saavedra, Francisco Javier Pérez-Fernández, Antonio Sala-Pérez

Abstract

In this manuscript we characterize the completeness of a normed space through the strong lacunary (N-theta) and lacunary statistical convergence (S-theta) of series. A new characterization of weakly unconditionally Cauchy series through N-theta and S-theta is obtained. We also relate the summability spaces associated with these summabilities with the strong p-Cesaro convergence summability space.

Characterizations of a Banach Space through the Strong Lacunary and the Lacunary Statistical Summabilities

Abstract

In this manuscript we characterize the completeness of a normed space through the strong lacunary (N-theta) and lacunary statistical convergence (S-theta) of series. A new characterization of weakly unconditionally Cauchy series through N-theta and S-theta is obtained. We also relate the summability spaces associated with these summabilities with the strong p-Cesaro convergence summability space.
Paper Structure (5 sections, 8 theorems, 31 equations)

This paper contains 5 sections, 8 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The statistical lacunary summability space
  4. The strong lacunary summability space
  5. Characterizations of the completeness of a Banach space

Key Result

Theorem 2.5

Let $X$ be a Banach space and $(x_k)$ a sequence in $X$. Notice that $S_\theta$ and $N_\theta$ are regular methods. Proof.

Theorems & Definitions (13)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Definition 2.7
  • Theorem 2.8
  • Theorem 3.1
  • Theorem 4.1
  • ...and 3 more