Absolute, perfect, and unconditional convergence of double series
type: Item
instance of: scholarly article
title: Absolute, perfect, and unconditional convergence of double series
zbMATH Open document ID: 0886.46009
DOI: 10.1007/BF02509457
author: M. I. Kadets
publication date: 1997-06-12T00:00:00Z
review text: A double series \(\sum x_{ij}\) of elements of a Banach space \(X\) is said to be convergent to \(s\), if each column \(\sum_i x_{ij}\) and each row \(\sum_j x_{ij}\) converges, and the ``rectangular'' sums of the double series converge to \(s\). \(\sum x_{ij}\) is called a) absolutely convergent, if \(\sum|x_{ij}|<\infty\), b) perfectly convergent, if \(\sum x_{ij}\alpha_i \beta_j\) converges for all \(\alpha_i=\pm 1\), \(\beta_j=\pm 1\), c) unconditionally convergent if \(\sum x_{\pi(i),\sigma(j)}\) converges for all permutations \(\pi\), \(\sigma\) of positive integers. There is given a necessary and sufficient condition for perfect convergence of \(\sum x_{ij}\). Moreover, perfect convergence is equivalent to unconditional convergence, and there exist double number series convergent perfectly but not absolutely.
reviewed by: Julian Musielak
Mathematics Subject Classification ID: 46B15
Mathematics Subject Classification ID: 40B05
Mathematics Subject Classification ID: 40J05
Mathematics Subject Classification ID: 40A10
zbMATH DE Number: 999679
zbMATH Keywords: double series
zbMATH Keywords: absolutely convergent
zbMATH Keywords: perfectly convergent
zbMATH Keywords: unconditionally convergent
MaRDI profile type: Publication
cites work: Q4000062
full work available at URL: https://doi.org/10.1007/bf02509457
OpenAlex ID: W2012703125
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 999679