The Exploration On Several Kinds Of Convexity And Smoothness In Locally Convex Spaces

In 1932,S.Banach defined the Banach spaces in?Theories of Operation Lineariness?for the first time,during decades of continuous exploration,development and improvement,Banach spaces theories have achieved a relatively complete results.As a direct extension of Banach spaces—locally convex spaces.In the past more than forty years,the theories of Banach spaces not only have drawn attention to many mathematicians,but also achieved a relatively perfect theoretical results,and these conclusions had practical value.For example,the drop properties,Asplund properties and Ekeland variational principles in locally convex spaces,etc.Although there are some scholars still study the locally p-convex spaces(where 0<p?1),because of the nonlinear structures in locally p-convex spaces.For now,It is rather difficult to introduce and study the convexity,smoothness and their dual properties in locally p-convex spaces.This paper based on the theoretical results of Banach spaces and locally convex spaces,try to appeal the convexity and smoothness of Banach spaces extend to locally convex spaces,discuss the characteristics,properties and the dual relationship between convexity and smoothness,obtain some results parallel to the Banach spaces geometry theories.The paper is divided into four chapters.Chapter 1:The basic knowledge.Chapter 2:We introduce the concepts and characteristics of k-extreme convexity and k-extreme smoothness in locally convex spaces.Under the condition of P-reflexivity,we consider the dual relationship in locally convex spaces.Chapter 3:We give the notions and characteristics of k-very extreme convexity and k-very extreme smoothness in locally convex spaces.Under the condition of P-reflexivity,we study the dual relationship in locally convex spaces.Chapter 4:We present the concepts and characteristics of mid-point locally k-very extreme convexity and mid-point locally k-very extreme smoothness in locally convex spaces.Under the condition of P-reflexivity,we discuss the dual relationship in locally convex spaces.