The Study On The Uniform Convexity And Differentiability Of The Continuous Function In Locally Space

In recent decades, the development of theoretical study on Banach space(or normed linear space) has been developing rapidly, on the contrary, the theoretical study on locally convex spaces, which is widely spread in Banach space, develops slowly. The scholars who study on the Drop property, Ekeland variational principle, the Asplund property and Differentiation Theory achieve great successes. Since the beginning of this century, the researches of locally convex space of differentiability Has also been a rapid development. In this paper, the author makes Gateaux differentiability and Frechet differentiability in Banach space into locally convex space,gives necessary and sufficient conditions about them and makes better Results. This paper is divided into four chapters:Let X be a real linear space, P a family of separated seminorms on X, (X, P) a dual, (X, Tp) a locally convex space generated by P.Chapter I:The knowledge of preparation.Chapter 2:This chapter gives a necessary and sufficient condition of Gateaux dif-ferentiability and Frechet differentiability of continuous gauge function in locally convex spaces.Chapter 3:This chapter gives a necessary and sufficient condition of continuous convex function differentiability in locally convex spaces.Chapter 4:This chapter gives a sufficient condition of the uniform convexity of locally convex spaces instead of Banach spaces.