Factorization of weakly continuous differentiable mappings
type: Item
instance of: scholarly article
title: Factorization of weakly continuous differentiable mappings
zbMATH Open document ID: 1184.46038
author: Raffaella Cilia
author: Joaquín M. Gutiérrez
published in: Bulletin of the Brazilian Mathematical Society. New Series
publication date: 2009-10-19T00:00:00Z
review text: For real Banach spaces \(X\) and \(Y\), let \(\mathcal{C}^1_{wbu}(X,Y)\) denote the space of Fréchet differentiable mappings \(f:X \to Y\) such that both \(f\) and its derivative \(f^\prime\) are weakly uniformly continuous on each bounded subset of \(X\). The authors show that \(f \in \mathcal{C}^1_{wbu}(X,Y)\) if and only if the mapping \(f^\prime:X \to \mathcal{K}(X,Y)\) (the space of compact operators \(X\to Y\)) is compact and uniformly continuous on bounded subsets of \(X\). Moreover, the authors establish a characterization of \(\mathcal{C}^1_{wbu}(X,Y)\) in terms of factorizations: \(f \in \mathcal{C}^1_{wbu}(X,Y)\) if and only if \(f = g\circ S,\) where \(S\) is a precompact operator into a normed space \(Z\) and \(g:Z \to Y\) satisfies certain differentiability and boundedness properties. As the authors note, the paper is partially motivated by work of the reviewer and \textit{J.\,B.\thinspace Prolla} [J.~Reine Angew.\ Math.\ 313, 195--216 (1980; Zbl 0413.41022)] and by their own contemporaneous work with \textit{G.\,Saluzzo} in [Proc.\ Am.\ Math.\ Soc.\ 137, No.\,5, 1743--1752 (2009; Zbl 1177.46057)].
Mathematics Subject Classification ID: 46E50
Mathematics Subject Classification ID: 46G05
zbMATH DE Number: 5617955
zbMATH Keywords: Fréchet differentiable mappinig
zbMATH Keywords: Gâteaux differentiable mapping
zbMATH Keywords: weakly uniformly continuous mapping
zbMATH Keywords: factorization of differentiable mappings
reviewed by: Richard M. Aron
MaRDI profile type: Publication
full work available at URL: s00574-009-0016-x
OpenAlex ID: W1964048986
cites work: Polynomial approximation of differentiable functions on Banach spaces.
cites work: Q3688914
cites work: Q4719886
cites work: Compact factorization of differentiable mappings
cites work: Schauder type theorems for differentiable and holomorphic mappings
cites work: Differential calculus in topological linear spaces
DOI: 10.1007/S00574-009-0016-X
description: scientific article; zbMATH DE number 5617955