On the Composition of Differentiable Functions

Mohammed Bachir (1) , Gilles Lancien (2)

1 Equations d'evolution
2 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)

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.

Domains: Functional Analysis [math.FA]

Mohammed Bachir, Gilles Lancien. On the Composition of Differentiable Functions. Canad. Math. Bull, 2003, 46 (4), pp.481-494. ⟨hal-01183277⟩