Application of a theorem of Grothendieck to vector measures
type: Item
instance of: scholarly article
title: Application of a theorem of Grothendieck to vector measures
zbMATH Open document ID: 0892.28009
author: Thiruvaiyaru V. Panchapagesan
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-07-15T00:00:00Z
review text: Let \(X\) be a quasi-complete locally convex space, \({\mathfrak R}\) a ring of subsets of a non-empty set and \(\Sigma({\mathfrak R})\) the Banach space of all bounded complex functions which are uniform limits of sequences of \({\mathfrak R}\)-simple functions. Mainly, a direct proof of the following result is given, without using the range theorem of I. Tweddle (1968) and an extension theorem of I. Kluvánek (1972): ``A continuous linear mapping \(u:\Sigma({\mathfrak R})\to X\) is weakly compact if and only if its representing measure is strongly additive''. This result subsumes the range theorems of Tweddle (1968) and Kluvánek (1972), and it also extends to quasi-complete locally convex spaces, Theorem VI.1.1 in ``Vector measures'' (1977; Zbl 0369.46039) by \textit{J. Diestel} and \textit{J. J. Uhl jun.}. Several others extensions (in different directions) are given. For instance, a generalization of the second part of Corollary VI.1.2 in the above-mentioned Diestel-Uhl's book, is also included.
Mathematics Subject Classification ID: 28B05
Mathematics Subject Classification ID: 46G10
zbMATH DE Number: 1115503
zbMATH Keywords: vector measure
zbMATH Keywords: locally convex space
zbMATH Keywords: ring of subsets
zbMATH Keywords: representing measure
zbMATH Keywords: range theorems
reviewed by: P. Jiménez Guerra
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5589
OpenAlex ID: W2089572920
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DOI: 10.1006/JMAA.1997.5589
description: scientific article; zbMATH DE number 1115503