On the interchange of series and some applications
type: Item
instance of: scholarly article
title: On the interchange of series and some applications
zbMATH Open document ID: 1089.46008
published in: Bulletin of the Belgian Mathematical Society - Simon Stevin
publication date: 2005-04-18T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.bbms/1093351381
review text: The basic matrix theorem of \textit{P. Antosik} and \textit{C. Swartz} [``Matrix methods in analysis'' (Lecture Notes in Mathematics 1113, Springer, Berlin etc.) (1985; Zbl 0564.46001)] says: If \((x_{ij})_{ij}\) is a matrix in a normed space \(X\) such that (1) for each \(j\in {\mathbb N},\) the sequence \((x_{ij})_i\) is convergent to some \(x_j\in X,\) and (2) for each infinite set \(M\subseteq {\mathbb N}\), there exists an infinite set \(P\subseteq M\) such that \((\sum_{j\in P} x_{ij})_i\) is a Cauchy sequence, then \((x_{ij})_i,\) \(j\in{\mathbb N},\) are uniformly convergent for \(j\in {\mathbb N}\) and \((x_j)_j\) converges to zero. In the paper under review, the authors improve the basic matrix theorem by using separation properties of natural families in \({\mathcal P}({\mathbb N})\) containing all finite subsets of \({\mathbb N}\) and satisfying weaker conditions than those in the above cited basic matrix theorem. In the case of Banach spaces \(X,\) the converse result is also proved and, as a corollary, a characterization of convergence of sequences in \(c_0(X)\) is given. Moreover, the authors generalize the Antosik-Swartz interchange theorem and use this result to characterize convergence of sequences in \(cs(X)\) and \(c(X),\) respectively, by the consideration of natural isometries from \(c_0(X)\) into these spaces. Another consequence of the proven interchange theorem is a version of \textit{C. Swartz}'s version of the Orlicz--Pettis theorem [Proc.\ Am.\ Math.\ Soc.\ 123, No.~2, 455--457 (1995; Zbl 0818.46010)] in the case of Banach spaces with a Schauder basis.
reviewed by: Johann Boos
Mathematics Subject Classification ID: 46B15
Mathematics Subject Classification ID: 46B25
Mathematics Subject Classification ID: 40A05
zbMATH DE Number: 2157935
zbMATH Keywords: basic matrix theorem
zbMATH Keywords: Orlicz-Pettis theorem
zbMATH Keywords: Antosik interchange theorem
zbMATH Keywords: spaces of sequences in a normed space
author: A. Aizpuru
author: Antonio Gutiérrez-Dávila
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2157935