Gâteaux differentiability of Lipschitz functions
Gâteaux differentiability of Lipschitz functions
This chapter presents the main results on Gâteaux differentiability of Lipschitz functions by recalling the notions of the Radon-Nikodým property (RNP) and null sets. The discussion focuses not only on the mere existence of points of Fréchet differentiability, but also, and often more important, on the validity of the mean value estimates. After considering the RNP of a Banach space, the chapter examines Haar and Aronszajn-Gauss null sets. It then analyzes the existence result for Gâteaux derivatives as well as the meaning of multidimensional mean value estimates. It also explains how, for locally Lipschitz maps of separable Banach spaces to spaces with the RNP, the condition for the validity of the multidimensional mean value estimate may be simplified.
Keywords: null sets, Gâteaux differentiability, Lipschitz function, Radon-Nikodým property, Fréchet differentiability, Banach space, Gâteaux derivative, multidimensional mean value
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