On the composition of differentiable functions

We prove that a Banach space X has the Schur property if and only if every X-valued weakly differentiable function is Frechet differentiable. We give a general result on the Frechet differentiability of f o 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 parallel to parallel to(lip) on various spaces of Lipschitz functions.