Banach Space Lipschitz Maps Differentiable,

This paper studies mainly characterizations of the differentiability of Lipschitz mappings on Banach spaces. We prove that for every Lipschitz mapping / from a Hilbert space H to Rn there exists a dense Gδ-subset F of H such that i) / is Prechet differentiable at every point of F ii) the derivative df of / is continuous on F; and moreover, iii) the set F can be chosen to be the Prechet differentiability point set of a Lipschitz convex function on H. Hence, it completely solves a long-standing problem concerning Prechet differentiability of Lipschitz mappings between Hilbert spaces, and also gives Lindenstrauss-Preiss' problem an affirmative answer; and further, it implicates a significant improvement of the Lebesgue-Rademacher theorem: every Lipschitz mapping between two finitely dimensional spaces is C1 off a null set of first category.The main tools used for obtaining the foregoing results are some approximation properties of Lipschitz functions by continuous A-convex functions and the convergence behavior of sequences of Lipschitz functions defined on Banach spaces. In this thesis, there are two results on these two topics as following:For the dual of a locally uniformly convexifiable Banach space, we show that every w*-lower semicontinuous Lipschitzian convex function can be uniformly approximated by a generically Frechet differentiable w*-lower semicontinuous monotone-nondecreasing Lipschitzian convex function sequence. On Hilbert spaces, we prove that every Lipschitz function can be uniformly approximated by a || . ||L0-bounded A-convex function sequence.Through giving a generalization of Choquet theorem to the dual of Lipschitz function spaces, we present a criterion for a sequences in Lipschitz function spaces to be weak convergence: Suppose that is a nonempty subset of a Banach space X and that f Lo( Ω), and suppose that (fn) L0(Ω ) is a bounded sequence such that Then We introduce the notion of Lipschitz maps with generically G-sequential compactnessproperty for studying the Gateaux differentiability of Lipschitz maps between Banach spaces, and prove that every Lipschitz mapping from a separable Hilbert space to Banach space Y with Schauder basis is Gateaux differentiable on a residual set if and only if the mapping has the generically G-sequential compactness property.