The Convexity In Every Direction Of Locally Convex Spaces And The Normal Structure Of Bounded Closed On Convex Sets

The main purpose of this thesis is to have a throughout research on convexities and smoothness in locally convex spaces and normal structures of bounded closed convex sets. These topics have attracted much attention of mathematicians for many years. In this thesis, for the first time, we introduce a definition of uniform convexity in every direction in locally convex spaces, and present several equivalent definitions. Also, we introduce a definition of uniform smoothness in every direction. Moreover, we prove a partial duality between the above two properties. At the last, we prove that the bounded closed convex sets have the normal structure in uniformly convex in every direction locally convex spaces.The thesis is divided into four chapters. The first one presents a brief discipline of the development in related fields and the main work in this paper; The second discusses convexities and its dual concepts in locally convex spaces and the duality between them; The third studies uniform convexities and uniform smoothness in locally convex spaces and some dualities between them. Furthermore, we prove that the bounded closed convex sets have normal structure in such spaces; In the fourth chapter, a definition of uniform convexity and several equivalent ones in every direction in locally convex space are given . Then we introduce for the first time the definition of uniform smoothness in every direction and prove the duality between the uniform convexity and the uniform smoothness in every direction. At the end, in terms of normal structures, we show the existence of fixed points of nonexpansive mappings.