Unconditionally Cauchy series and Cesàro summability
type: Item
instance of: scholarly article
title: Unconditionally Cauchy series and Cesàro summability
zbMATH Open document ID: 1114.46009
author: N. E. Zubov
published in: Journal of Mathematical Analysis and Applications
publication date: 2006-12-07T00:00:00Z
review text: Let \(X\) be a normed linear space, \(\sum x_{j}\) a (formal) series in \(X\) and \( \mathcal{S}(\sum x_{j})\) \([\mathcal{S}_{w}(\sum x_{j})]\) be the subspace of \( l^{\infty }\) consisting of those sequences \(\{t_{j}\}\) such that the series \(\sum_{j=1}^{\infty }t_{j}x_{j}\) converges \([weakly]\) with the sup norm. It has been shown that the space \(X\) is complete iff for every weakly unconditionally convergent series \(\sum x_{j}\) in \(X\) the space \(\mathcal{S} (\sum x_{j})\) \([\mathcal{S}_{w}(\sum x_{j})]\) is complete. Similarly, the space \(X\) is barrelled iff for each series \(\sum x_{j}^{\prime }\) in \( X^{\prime }\) which is such that \(\mathcal{S}_{w\ast }(\sum x_{j})=l^{\infty } \) is weakly unconditionally Cauchy [\(\mathcal{S}_{w\ast }(\sum x_{j})\) is defined as above using the weak* topology]. In this paper the authors extend these results by replacing the usual convergence of the series by Cesaro convergence.
Mathematics Subject Classification ID: 46B15
Mathematics Subject Classification ID: 40G05
zbMATH DE Number: 5078045
zbMATH Keywords: (weakly) unconditionally Cauchy series
zbMATH Keywords: Cesàro-convergence
zbMATH Keywords: weak (and weak-*) Cesàro-convergence
zbMATH Keywords: Orlicz-Pettis theorem
zbMATH Keywords: completeness and barrelledness
reviewed by: Charles Swartz
MaRDI profile type: Publication
full work available at URL: j.jmaa.2005.11.071
OpenAlex ID: W1983666179
cites work: Q4944765
cites work: Summing Boolean algebras
cites work: On bases and unconditional convergence of series in Banach spaces
cites work: Q3329946
cites work: Q3839703
cites work: On Relationships Amongst Certain Spaces Of Sequences In An Arbitrary Banach Space
cites work: Characterizations of completeness of normed spaces through weakly unconditionally Cauchy series
cites work: Summability and substructures of \(2^{N}\)
cites work: A general Hahn-Schur theorem
cites work: An automatic adjoint theorem and its applications
cites work: A summation theorem and its applications
cites work: A general Orlicz-Pettis theorem
DOI: 10.1016/J.JMAA.2005.11.071
description: scientific article; zbMATH DE number 5078045