Vol. 77, No. 2, 1978

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Gaussian null sets and differentiability of Lipschitz map on Banach spaces

Robert Ralph Phelps

Vol. 77 (1978), No. 2, 523–531

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.

Mathematical Subject Classification 2000
Primary: 46G05
Secondary: 58C20
Received: 2 September 1977
Published: 1 August 1978
Robert Ralph Phelps