HADAMARD DIFFERENTIABILITY VIA GATEAUX DIFFERENTIABILITY

Abstract

Let f be a mapping from a separable Banach space to a Banach space. We prove that, except for a sigma-directionally porous set, f is Hadamard differentiable at those points, at which f is Lipschitz and Gateaux differentiable.

As a consequence we obtain that an everywhere Gateaux differentiable mapping from an Euclidean space to a Banach space is Frechet differentiable except for a nowhere dense sigma-porous set.

Keywords

• Hadamard differentiability
• Frechet differentiability
• sigma-porous set
• sigma-directionally porous set
• Stepanoff theorem
• Aronszajn null set