In this note we introduce and
briefly study the notion of a Gaussian null set in a real separable Banach space
E. As a corollary to recent work of Aronszajn we then show that a locally
Lipschitz mapping from E into a Banach space with the Radon-Nikodym
property is Gateaux differentiable outside of a Gaussian null set. This is
an infinite dimensional generalization of Rademacher’s classical theorem
that such mappings from Rn to Rm are differentiable almost everywhere
(Lebesgue). This approach will be compared with another generalization of
Rademacher’s theorem due independently to Christensen and Kaier and to
Mankiewicz.
