The Study On Some Kinds Of Convexity In B-Z-Spaces

In recent decades, the research on the theory of Banach spaces geometry has been rapidly developed, especially in strict concexity and smoothness, k-strict and k-smoothness, k-uniform coxvexity and k-uniform smoothness, k-strong convexity and k-strong smoothness, k-weak convexity and k-weak smoothness that research of the theory of convexity and smoothness is ideal, and some results are obtained. However, as the theory of the Banach spaces directly extended to the B-Z-spaces, the results are relatively slow. In this paper, we introduce the concepe of weak convexity, k-weak convexity, k-weak smoothness, k-strong convexity and k-smoothness of B-Z-spaces. Related Convexity and Smoothness of Banach Spaces clever promotion to the B-Z-spaces, and obtain some good results. This paper is divided into three chapters.Chapter 1: The knowledge of preparation.Chapter 2: In this chapter, we introduce the concept of weak convexity, smoothness and uniform smoothness of Z-spaces. It is proved that the uniform convexity and uniform smoothness are dual in Z-spaces, at the same time, the k-weak convexity and the k-weak smoothness of Banach spaces are extended to B-Z-spaces, and some equivalent conditions of k-weak convexity and k-weak smoothness are obtained.Chapter 3: In this chapter, we introduce the concept of drop property, weakly drop property, strong convexity and k-strong convexity of B-Z-spaces. They are the popularize of Banach spaces to B-Z-spaces. It is proved that the drop property and weak drop property are also set up in B-Z-spaces.