The Study On The Several Kinds Of Convexity And Smoothness In Banach Spaces

In this thesis, we mainly explore the convexity and smoothness of Banach spaces, and discuss their relations of some convexity and smoothness, which have been known. In the process of my study, I have obtained some productive results.This paper consists of three parts.In chapter one, the conceptions of K-very extremely convexity and K-very extremely smoothness are introduced as generalizations of very extremely convexity and very extremely smoothness. It is shown that K-very extremely convex (respectively. K-very extremely smooth) spaces lies strictly between K-uniformly extremely convex and K-very convex (respectively. K-uniformly extremely smooth and K-very smooth) spaces. There is no inclusion relations with the class of K-very extremely convex and K-strongly convex (respectively. K-very extremely smooth and K-strongly smooth) spaces. It is proved that K-very extremely convex (respectively. K-very extremely smooth) spaces include (K+1)-very extremely convex (respectively. (K+1)-very extremely smooth) spaces. However, the converse inclusion is not necessarily true. The some necess -ary and sufficient conditions for a Banach space to be K-very extremely convex spaces or K-very extremely smooth spaces are obtained. In chapter two, we show that Xian Jun and Hu Changsong's notion of K-extremely convexity and He Renyi's notion of K-extremely convexity are not the duality notion of K-extremely smoothness. We properly introduce the notion of K-extremely convexity, and show that the K-extremely convexity which we introduce is just the notion of duality of Xian Jun and Hu Changsong's K-extremely smoothness and lies strictly between the Xian Jun and Hu Changsong's notion of K-extremely convexity and He Renyi's notion of K-extremely convexity. In addition, we obtain some property of this kind of convexity and study the relation between that with various convexity.In chapter three, we introduce the notion of average uniformly smooth spaces, which is the dual of average uniformly convex spaces. i.e.,X *is average uniformly convex space if and only if X is average uniformly smooth space, and X *is average uniformly smooth spaces if and only if X is average uniformly convex space. Also, we show that the relations among the average uniform smoothness and other smoothness.