Locally solid topologies on vector valued function spaces
type: Item
instance of: scholarly article
title: Locally solid topologies on vector valued function spaces
zbMATH Open document ID: 0912.46028
author: Krzysztof Feledziak
author: Marian Nowak
published in: Collectanea Mathematica
publication date: 1999-05-03T00:00:00Z
full work available at URL: https://eudml.org/doc/40751
review text: Let \(M\) denote the set of strongly measurable functions with values in a Banach space. A set \(E\subseteq M\) is called solid if \(y\in E\) and \(\| x(t)\|\leq\| y(t)\|\) (a.e.) implies \(x\in E\). If \(E\) is a topological space and \(0\) has a base of solid neighborhoods, the topology is called locally solid. In this paper, locally solid topologies are characterized. Moreover, many useful properties which are known in the normed case (for ideal spaces/Köthe spaces) are studied, e.g.: Continuity of the embedding of \(E\) into \(M\); analogues of the convergence theorems of Lebesgue and Vitali; the finest topology in Orlicz-Bochner spaces for which such theorems hold is characterized. The proofs reduce the problems to the scalar case which was studied in e.g. by \textit{W. Wnuk} [Bull. Pol. Acad. Sci., Math. 34, 413-416 (1986; Zbl 0605.46008)] and \textit{M. Novak} [ibid. 32, 439-445 (1984; Zbl 0584.46018)].
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 46E40
Mathematics Subject Classification ID: 46A80
Mathematics Subject Classification ID: 46A40
zbMATH DE Number: 1186790
zbMATH Keywords: vector valued function
zbMATH Keywords: Köthe function space
zbMATH Keywords: Banach function space
zbMATH Keywords: ideal space
zbMATH Keywords: Orlicz space
zbMATH Keywords: topological vector space
zbMATH Keywords: locally solid space
zbMATH Keywords: modular space
zbMATH Keywords: strongly measurable functions with values in a Banach space
zbMATH Keywords: continuity of the embedding
zbMATH Keywords: convergence theorems of Lebesgue and Vitali
zbMATH Keywords: finest topology in Orlicz-Bochner spaces
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1186790