K-convexity, K-smoothness, Near-convexity And Near-smoothness In Banach Spaces

In this paper,we discussed K-Uniform Convexity,K-Uniform Smoothness and the dual relationship between them. At the same time we discussed Near-Uniform Convexity,Near-Uniform Smoothness and the dual property between them. Using the unit ball's slice,we described them. Then get the relationship between K-Convexity and Near-Convexity and the relationship between K-Smoothness and Near-Smoothness. So we simplified the proposition 'K-uniform convexity is Near-uniform convexity',which had been proved by Yu Xin-tai in reference [39],and got some new results.We introduced some new concepts while we were discussing all convexity and smoothness,and we proved some equal propositions. In section 1.1,on the basis of k-dimension volume,we introduced the notion of KDC for the first time. In section 1.2,using the unit ball's slice,we described it. Then proved that it is one kind of convexity between KUC and KSC,and as one concept,KDC is the dual concept of KSS better than KSC is. At the same time,we proved that the concepts of KSC,which were defined by Suyalatu in reference [27] and by He Renyi in reference [13],are equal. From some definitions and some theorems,we also got some results as following:After discussing the relationship between KUC and KUS,we proved that the concept of LKUC is the dual notion of LKUS. Then we got an important corollary In chapter 2,we discussed Near-Convexity and Near-Smoothness. Using the non-compact measure,we described them. Then using the unit ball's slice,we got such results:And we proved that LNUC is dual notion of LNUS,Drop property is the dual notion of NSS. At last we gave the relationship between NS'C and NSS,and the relationship between NS'C and NS.Because we had described the notions of all convexities and smoothness by using the unit ball's slice,in chapter 3,we derived the relationship between K-Convexity and Near-Convexity,and the relationship between K-Smoothness and Near-Smoothness directly.In chapter 4,we discussed Uniformly Extreme Smoothness and Very Extreme Smoothness. And proved the proposition 'X is Uniformly Extreme Smoothness if and only if X is Strong Smoothness and reflective. So the notion of Uniformly Extreme Smoothness,whichwas introduced in reference [19],is one kind of smoothness that between Uniform Smoothness and Strong Smoothness. We also proved the relationship between Extreme Smoothness and Local Near-Smoothness.