On the Composition of Differentiable Functions
Résumé
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 $\|.\|_{lip}$ on various spaces of Lipschitz functions.