Subseries convergence in the space of functions of bounded \(\phi\)- variation
type: Item
instance of: scholarly article
title: Subseries convergence in the space of functions of bounded \(\phi\)- variation
zbMATH Open document ID: 0611.46015
DOI: 10.1007/BF03322654
published in: Results in Mathematics
publication date: 1985-01-01T00:00:00Z
review text: An infinite series \(\sum x_ i\) in an F-space X is said to be ''perfectly bounded'' if the set \(\{\sum \epsilon_ ix_ i:\epsilon_ i=0\) or \(1\}\) is bounded. It is called ''perfectly convergent'' if every subseries \(\sum \epsilon_ ix_ i\) is convergent, i.e., if the given series is unconditionally convergent. In this paper, conditions are determined for a series of elements in the space of functions of bounded \(\phi\)-variation to have the property that every perfectly bounded series is necessarily perfectly convergent. For example, a classical result due to Orlicz shows that in a weakly sequentially complete Banach space, every weakly bounded series has this property.
Mathematics Subject Classification ID: 46B15
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 3987961
zbMATH Keywords: subseries convergence
zbMATH Keywords: perfectly bounded
zbMATH Keywords: perfectly convergent
zbMATH Keywords: space of functions of bounded \(\phi \)-variation
author: Jarosław Ciemnoczołowski
author: Władysław Orlicz
reviewed by: Robert M. Young
MaRDI profile type: Publication
cites work: Q3681373
cites work: On generalized variations (II)
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cites work: On generalized variations (I)
cites work: On modular spaces
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cites work: Linear operations in Saks spaces (I)
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cites work: General inequalities of Stieltjes integrals and the convergence of Fourier series
full work available at URL: https://doi.org/10.1007/bf03322654
OpenAlex ID: W1972831721
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description: scientific article; zbMATH DE number 3987961