Journal of Convex Analysis 21 (2014), No. 3, 703--713
Copyright Heldermann Verlag 2014
Gâteaux and Hadamard Differentiability via Directional Differentiability
Ludek Zajícek
Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic
zajicek@karlin.mff.cuni.cz
[Abstract-pdf]
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.
Keywords: Gateaux differentiability, Hadamard differentiability, directional derivatives, Hadamard directional derivatives, sigma-directionally porous set, pointwise Lipschitz mapping.
MSC: 46G05; 26B05, 49J50
[ Fulltext-pdf (131 KB)] for subscribers only.