Exploration On Several Kinds Of Convexity And Smoothness In Several Locally Spaces

The research of convexity and smoothness of Banach space is an important part of the geometric theory of Banach space. As uniformly convex is introduced by J.A.Clarkon, the research of convexity and smoothness of Banach space is in-creasingly popular domestic and international. As so far, it has been perfected, but some known convexity and smoothness of Banach space has not been spread to lo-cally convex separated space. In this article, K average strong convexity, K average strong smoothness, K very extreme convexity and K very extreme smoothness of Banach space is respectively spread to locally convex separated space, and some relevant properties are obtained.In chapter1, some basic knowledge and notations of locally convex separated space are introduced.In chapter2, the concept of affine strict convexity and affine smoothness is in-troduced to locally convex separated space, and is spread to K affine strict convexity and K affine smoothness respectively. The duality of K affine strict convexity and K affine smoothness is obtained.In chapter3, very extreme convexity and very extreme smoothness of Banach space are spread to locally convex separated space. K very extreme convexity and K very extreme smoothness, also duality of them, are obtained.In chapter4, average strong convexity and average strong smoothness of Banach are spread to locally convex separated space. K average strong convexity and K average strong smoothness, also duality of them, are obtained.In chapter5, Summarize this article, and point out some unresolved issues.