G\^ ateaux and Hadamard differentiability via directional differentiability

Abstract

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 $f_{H+}^{\prime }(x,u)$ 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.