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On the Composition of Differentiable Functions

Published:2003-12-01
Printed: Dec 2003
• M. Bachir
• G. Lancien
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Abstract

We prove that a Banach space $X$ has the Schur property if and only if every $X$-valued weakly differentiable function is Fr\'echet differentiable. We give a general result on the Fr\'echet differentiability of $f\circ T$, where $f$ is a Lipschitz function and $T$ is a compact linear operator. Finally we study, using in particular a smooth variational principle, the differentiability of the semi norm $\Vert \ \Vert_{\lip}$ on various spaces of Lipschitz functions.
 MSC Classifications: 58C20 - Differentiation theory (Gateaux, Frechet, etc.) [See also 26Exx, 46G05] 46B20 - Geometry and structure of normed linear spaces