A Banach-Zarecki theorem for functions with values in Banach spaces
type: Item
instance of: scholarly article
title: A Banach-Zarecki theorem for functions with values in Banach spaces
zbMATH Open document ID: 1310.28009
author: Sokol Bush Kaliaj
published in: Monatshefte für Mathematik
publication date: 2014-12-03T00:00:00Z
review text: This paper gives a vector valued version of the classical Banach-Zarecki theorem, which establishes that a function \(F:[a,b] \to \mathbb R\) is strongly absolutely continuous if and only if \(F\) is of bounded variation, continuous on \([a,b]\) and satisfies the property (N), that is, for each Lebesgue measurable subset \(E\) of \([a,b]\), its Lebesgue measure \(\lambda(E)=0\) implies that \(\lambda(F(E))=0\). The vector valued version, that is proved in the paper, states the following (Theorem 2.1): Let \(X\) be a Banach space and let \(F:[0,1] \to X\) be a function with compact range. The following assertions are equivalent. {\parindent =6mm \begin{itemize}\item[(i)] \(F\) is strongly absolutely continuous with respect to the norm. \item[(ii)] \(F\) is of bounded variation, weak continuous and satisfies the weak property (N). \end{itemize}} A function \(F\) has the weak property (N) is for each \(x^* \) in the dual of \(X\), we have that the function \(x^* \circ F\), has property (N).
reviewed by: E. A. Sánchez-Pérez
Mathematics Subject Classification ID: 28B05
Mathematics Subject Classification ID: 26A45
Mathematics Subject Classification ID: 26A46
Mathematics Subject Classification ID: 26A48
Mathematics Subject Classification ID: 46G10
Mathematics Subject Classification ID: 26E20
zbMATH DE Number: 6376632
zbMATH Keywords: Banach-Zarecki theorem
zbMATH Keywords: Banach space
zbMATH Keywords: strong absolute continuity
zbMATH Keywords: bounded variation
zbMATH Keywords: property (N)
zbMATH Keywords: Minkowski functional
MaRDI profile type: Publication
full work available at URL: s00605-014-0609-3
OpenAlex ID: W2111150528
cites work: On Denjoy type extensions of the Pettis integral
cites work: The Banach-Zarecki theorem for functions with values in metric spaces
cites work: Q3254057
cites work: Real functions - current topics
cites work: Q5565773
cites work: Q3999910
cites work: The differentiability of Banach space-valued functions of bounded variation
cites work: Q3775752
cites work: Q4273174
cites work: Q4788811
cites work: Q5620935
cites work: Q5708391
cites work: Pettis integral and measure theory
cites work: On Absolutely Continuous Functions
DOI: 10.1007/S00605-014-0609-3
Recommended article: The Banach-Zarecki theorem for functions with values in metric spaces
Recommended article: A generalization of the Banach Zarecki theorem
Recommended article: Paley-Wiener-type theorem for functions with values in Banach spaces
Recommended article: Tauberian theorems for generalized functions with values in Banach spaces
Recommended article: An implicit function theorem in Banach spaces
Recommended article: The Banach-Saks property for function spaces
Recommended article: Q4246523
Recommended article: Absolutely continuous functions with values in a Banach space
Recommended article: Q3338757
Recommended article: Q3111332
description: scientific article; zbMATH DE number 6376632