Frechet differentiability via partial Frechet differentiability

TL;DR

This paper investigates the conditions under which partial Fréchet differentiability implies full Fréchet differentiability in Banach spaces, revealing that certain sets of points where this does not hold are small in a topological or measure sense.

Contribution

It establishes that in Banach spaces with Asplund properties, the set of points where partial but not full Fréchet differentiability occurs is small, and extends results to Lipschitz functions between separable Banach spaces.

Findings

Set of points with partial but not full Fréchet differentiability is first category or σ-upper porous.

In Lipschitz mappings, points with Gâteaux but not Fréchet differentiability form a σ-upper porous set.

Results generalize differentiability properties in Banach spaces with Asplund and separable conditions.

Abstract

Let $X_1, \dots, X_n$ be Banach spaces and $f$ a real function on $X=X_1 \times\dots \times X_n$. Let $A_f$ be the set of all points $x \in X$ at which $f$ is partially Fr\' echet differentiable but is not Fr\' echet differentiable. Our results imply that if $X_1, \dots, X_{n-1}$ are Asplund spaces and $f$ is continuous (resp. Lipschitz) on $X$, then $A_f$ is a first category set (resp. a $\sigma$-upper porous set). We also prove that if $X$, $Y$ are separable Banach spaces and $f:X \to Y$ is a Lipschitz mapping, then the set of all points $x \in X$ at which $f$ is G\^ ateaux differentiable, is Fr\' echet differentiable along a closed subspace of finite codimension but is not Fr\' echet differentiable, is $\sigma$-upper porous. A number of related more general results are also proved.