The Study On The Some K Convexity And K Smoothness Of Banach Spaces

In this thesis, we mainly explore the K convexity and K smoothness of Banach spaces, and discuss their relations of some K convexity and K smoothness, which have been known. In the process of my study, I have obtained some productive results.This paper consists of four parts.Chapter one the basic definitions and necessary lemmas posed in the paper.Chapter two the generalization of uniformly extremely convex space and uniformly extremely smooth space.In chapter two, we introduce the definitions of K uniformly extreme convexity and K uniformly extreme smoothness, and give their properties and some characteristic descriptions, and establish their dual relation .At the same time, the relations of K uniformly extreme convexity (resp.K uniformly extreme smoothness) and some K convexity (resp.K smoothness),which have been known, are discussed in this paper.The concepts of K uniformly extreme convexity and K uniformly extreme smoothness given in the thesis are the generalization of uniformly extreme convexity and uniformly extreme smoothness, especially in the case of k=1, the results in chapter two have contained all the results in uniformly extremely convex space and uniformly extremely smooth space.Chapter three The generalization of even space and strongly evenspaceIn chapter three, we introduce K even space and K strongly even space, and give their properties and some characteristic descriptions. On the basis of them, we obtain some important results as follows: (ï¿¡+1) strongly even space =># strongly even space => K even space => (AT-l)even space. Up to now, K strongly even space is very bad Banach space in convexity. The concepts of K strongly even space and K even space given in the paper are the generalization of strongly even space and even space, especially in the case of k=\, the results in chapter three have contained all the results in strongly even space and even space. In addition, we also introduce the concepts of strongly even module, K strongly even module, and give one sufficient and necessary condition for them respectively.Chapter four K uniformly smooth space and its applications.Aiming at three different definitions of K uniformly smooth space appeared in 1997, we prove these definitions are equivalent in essence though their forms are different. As the applications of ï¿¡ uniformly smooth space, we obtain two important properties of them, and prove that the properties are necessary and sufficient conditions for a Banach space to be K uniformly smooth space as well.