Factorization of weakly continuous differentiable mappings
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)].
factorization of differentiable mappings
scientific article; zbMATH DE number 5617955