Convex Komlós sets in Banach function spaces
type: Item
instance of: scholarly article
title: Convex Komlós sets in Banach function spaces
zbMATH Open document ID: 1210.46021
author: Jerry B. Day
author: Chris Lennard
published in: Journal of Mathematical Analysis and Applications
publication date: 2010-03-29T00:00:00Z
review text: Komlós's theorem establishes that, for any bounded sequence in \(L^1[0,1]\), there is a subsequence and a function \(g \in L^1[0,1]\) such that, for any further sequence, its Cesàro means converge to \(g\) (see [\textit{J.\,Komlós}, Acta Math.\ Acad.\ Sci.\ Hung.\ 18, 217--229 (1967; Zbl 0228.60012)]). A sort of converse of this result that states that every convex subset of \(L^1(\mu)\) satisfying the conclusion of Komlós Theorem is norm bounded was proved by \textit{C.\,Lennard} [Pac.\ J.\ Math.\ 159, No.\,1, 75--85 (1993; Zbl 0739.46017)]. In this paper, these results -- Komlós's theorem and the Lennard converse -- are extended to the class of Banach function spaces over \(\sigma\)-finite measure spaces that are finitely integrable and satisfy the Fatou property.
reviewed by: E. A. Sánchez-Pérez
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 28A20
Mathematics Subject Classification ID: 46B25
Mathematics Subject Classification ID: 46A55
zbMATH DE Number: 5687686
zbMATH Keywords: Komlós
zbMATH Keywords: Banach function space
zbMATH Keywords: Fatou property
zbMATH Keywords: \(L^p\)-space
zbMATH Keywords: Cesàro convergence
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.12.040
OpenAlex ID: W2080990722
cites work: A general strong law
cites work: The uniform Kadec–Klee property for Orlicz–Lorentz spaces
cites work: A generalization of a problem of Steinhaus
cites work: A converse to a theorem of Komlós for convex subsets of \(L_ 1\)
cites work: On a problem of Steinhaus
DOI: 10.1016/J.JMAA.2009.12.040
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description: scientific article; zbMATH DE number 5687686