G\^ ateaux and Hadamard differentiability via directional differentiability

TL;DR

This paper establishes that under certain conditions, the existence of directional derivatives in a dense set of directions implies Hadamard differentiability outside a small porous set, generalizing previous results in Banach space theory.

Contribution

It extends and improves existing theorems by showing that directional derivatives in a dense set of directions imply Hadamard differentiability, even when the set of directions is not the entire space.

Findings

Hadamard differentiability holds outside a sigma-directionally porous set.

Generalizes Ioffe's result to broader settings.

Connects directional derivatives with Gâteaux and Hadamard 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 $f'_{H+}(x,u)$ 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 $Y = \R$. 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.