Differentiability Of Lipschitz Mappings And Embeddings Of Convex Sets Of Banach Spaces

This paper is committed to two different areas of the research of functional analysis and combines them - Lipschitz embedding of Banach space and differentiability of Lipschitz mapping.We have adopted a new approach to achieve some surprising conclusions such as "The Lipschitz embedding from any convex set into a Banach space with RNP is equivalent to linear embedding","If a separated Banach spaceâ…©with RNP is Lipschitz embedded into c₀ with Lipschitz mapping T,then T(â…©) can not contain a convex subset of which the linear span is an infinite dimensional space " Our basic approach is to give a precise characterization of nonempty closed convex set of Banach space through the research of non-support points and we proof that the closed convex set of separated Banach space has "non-zero" measure if and only if it has non-support point(Chapter 2);then localize the classic G(?)teaux differentiability theorem,(chapter 3)that is,the Lipschitz mapping from any convex set into a Banach space with RNP is G(?)teaux differentiable almost everywhere;We apply it to the linear embedding about convex subset of Banach space(Chapter 3) and consider the coarse embedding(Chapter 4);Finally(Chapter 5) discuss the characterization of a nonempty closed convex set with super-CCP.