Further Exploration On Some K-smoothness Of Banach Spaces

In the recent years, the studying on Banach space geometric theory has been developed rapidly. Up to now, the explorations on convexity, smoothness, differentiability, roughness, drop-property and the convergence of Banach space has been perfect relatively, but the generalization of some known convexity and smoothness of Banach spaces has not been deeply studied. Therefore, in this paper, I mainly explore the three kinds of k-smoothness in Banach spaces, and divide the paper three chapters.Chapter one: In the chapter, we studied the k-smoothness of second dual space of X by using the local reflexive principle, and obtained the some characteristic of them. Chapter two: We studied the k- very smooth space by using the localreflexive principle, and obtained the following result: X is k-very smooth space if and only if for any x∈S(X), there holds dim S_x =r≤kand x|^ is r-smooth point of X⁺⁺ . Also, we obtained the some new characteristic of k-very smooth Banach spaces.Chapter three: We studied the k- very smooth spaces , and obtained some properties of them. Also, we obtained the relationships between k-extremely smoothness and various types k-smoothness. In addition, we prove that if X is weakly local k-uniformly convex spaces, then for each point x∈S (X),f∈S_x are the k -smooth points of X^* .