On the Composition of Differentiable Functions

http://dx.doi.org/10.4153/CMB-2003-047-2
Canad. Math. Bull. 46(2003), 481-494

  Published:2003-12-01
 Printed: Dec 2003

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.