The Study On Several Convexity And Smoothness Of Locally Convex Space

In recent years, theoretical research of the Banach space has been got the rapid development, but as a Banach space directly promoting the study of the theory of locally convex space,it is relatively slow. In the study of the theory of the locally convex space, drops of locally convex space, Ekland variational principle, the nature of Asplund and differential theory research are almostly ideal,these properties all have a stronger application.In this article, the theory of Banach space method is used to introduce several convexity and smoothness of locally convex space, discussing their characteris-tics, nature and the relationship between each other, and it has got some good results. Since the weak compactness of locally convex space is different from weak sequence, how to draw out the smoothness and convexity of locally convex space is the key to deal with the relationship between weakly compact and weak sequence.This is the features and innovation of this article.This article introduced in locally convex space k-very convexity;K-extreme smoothness;K-very strictly convex;K-strict smooth;Uniform convexity;Uniform smoothness. The related convexity and smoothness in Banach space to locally convex space, Their necessary and sufficient conditions are given, made some good results, this article is divided into four chapters.Chapter 1:The knowledge of preparation.Chapter 2:This chapter gives sufficient and necessary conditions to the lo-cally convex space k-extremely convexity and k smoothness.Then K-extremely convex Banach space and k-the very definition of smoothness are promoted to locally convex space.Chapter 3:This chapter gives sufficient and necessary conditions to the lo-cally convex space K-strict convexity and K smoothness-very strict. Then it promotes the Banach space K-very strict convexity and K smoothness-very strict definition to locally convex space.Chapter 4:This chapter gives a locally convex space uniform convexity and uniform smoothness of necessary and sufficient condition. Then it promotes the Banach space uniform convexity and uniform smoothness very definition to locally convex space.