Iterated series and the Hellinger-Toeplitz theorem
type: Item
instance of: scholarly article
title: Iterated series and the Hellinger-Toeplitz theorem
zbMATH Open document ID: 0766.40002
DOI: 10.5565/PUBLMAT_36192_12
published in: Publicacions Matemàtiques
publication date: 1993-01-16T00:00:00Z
full work available at URL: https://eudml.org/doc/41161
review text: The paper deals with the problem of interchanging summations in iterated double series. Simple cases when this can be done are when all terms are non-negative (in which case sums may be infinite) or when the double sum converges absolutely. In most other cases sufficient conditions are difficult to verify. A condition by Antosik involves only the iterates, and guarantees the existence of the double limit, and hence the equality of the two iterated series. A vector-valued version is presented, and is proved by using a vector-valued generalization of the Schur lemma on weakly convergent series in \(\ell^ 1\). Of interest is here (also in the scalar case) that the condition in the hypothesis (involving only iterated series) sometimes is easily checked. An example of such a condition is shown in the proof of a Hellinger-Toeplitz result. The classical Hellinger-Toeplitz theorem states that any matrix mapping of \(\ell^ 2\) into \(\ell^ 2\) is norm continuous. The generalization in the present paper deals with vector spaces \(E\), \(F\) of sequences with values in Hausdorff topological vector spaces \(X\), \(Y\), such that \((X,Y)\) has the Banach-Steinhaus property. The result proved asserts that a matrix transformation \(A\in(E,F)\) is continuous with respect to certain weak topologies on \(E\) and \(F\) under appropriate conditions on \(E\) and \(F\).
reviewed by: Haakon Waadeland
Mathematics Subject Classification ID: 40A05
Mathematics Subject Classification ID: 47A05
zbMATH DE Number: 94213
zbMATH Keywords: Antosik condition
zbMATH Keywords: double series
zbMATH Keywords: Schur lemma
zbMATH Keywords: Hellinger-Toeplitz theorem
zbMATH Keywords: Banach-Steinhaus property
author: Charles Swartz
MaRDI profile type: Publication
OpenAlex ID: W1990513847
description: scientific article; zbMATH DE number 94213