Completeness of \(L^1\)-spaces for measures with values in complex vector spaces
type: Item
instance of: scholarly article
title: Completeness of \(L^1\)-spaces for measures with values in complex vector spaces
zbMATH Open document ID: 0913.46036
author: Werner J. Ricker
author: Antonio Fernández
author: Francisco Naranjo
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-11-29T00:00:00Z
review text: A direct proof of a theorem is given stating that if \(X\) is a complex Fréchet space and \(\nu:\Sigma\to X\) is a vector measure, then the space \(L^1(\nu,X)\) is \(\tau(\nu)\)-complete and thus it is a Fréchet space. Moreover, it is proved that if \(X\) is a complex sequentially complete, locally convex Hausdorff space, \(\nu: \Sigma\to X\) -- a vector measure, then \(L^1(\nu, X)\) is complete if and only if \(\nu\) is a closed measure.
Mathematics Subject Classification ID: 46E40
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 46G10
Mathematics Subject Classification ID: 46A04
zbMATH DE Number: 1215663
zbMATH Keywords: completeness
zbMATH Keywords: complex Fréchet space
zbMATH Keywords: vector measure
zbMATH Keywords: sequentially complete, locally convex Hausdorff space
reviewed by: Julian Musielak
MaRDI profile type: Publication
full work available at URL: jmaa.1998.5956
OpenAlex ID: W2026492149
cites work: Rybakov's theorem for vector measures in Fréchet spaces
cites work: Integration with respect to vector measures
cites work: Vector measures and integration in non-complete spaces
cites work: On Boolean Algebras of Projections and Scalar-Type Spectral Operators
cites work: Q4881908
cites work: Q4405392
DOI: 10.1006/JMAA.1998.5956
description: scientific article; zbMATH DE number 1215663