Gâteaux and Hadamard Differentiability via Directional Differentiability

L. Zajícek

Journal of convex analysis, Tome 21 (2014) no. 3, pp. 703-713

Résumé

Let $X$ be a separable Banach space, $Y$ a Banach space and $f: X \to Y$ an arbitrary mapping. Then the following implication holds at each point $x\in X$ except a $\sigma$-directionally porous set:\ If the one-sided Hadamard directional derivative $f'_{H+}(x,u)$ exists in all directions $u$ from a set $S_x \subset X$ whose linear span is dense in $X$, then $f$ is Hadamard differentiable at $x$. This theorem improves and generalizes a recent result of A. D. Ioffe, in which the linear span of $S_x$ equals $X$ and $Y = \mathbb{R}$. An analogous theorem, in which $f$ is pointwise Lipschitz, and which deals with the usual one-sided derivatives and G\^ ateaux differentiability is also proved. It generalizes a result of D. Preiss and the author, in which $f$ is supposed to be Lipschitz.

Détail

Classification : 46G05, 26B05, 49J50
Mots-clés : Gateaux differentiability, Hadamard differentiability, directional derivatives, Hadamard directional derivatives, sigma-directionally porous set, pointwise Lipschitz mapping