The Study On The Some Convexity And Smoothness Of Orlicz Spaces

In this thesis, we mainly explore the convexity and smoothness of Orl -icz spaces, and discuss their relations of some convexity and moothness, which have been known. In the process of my study, I have obtained some productive results.This paper consists of four parts.Chapter one:In this chapter, the basical definitions and necessary lemmas are introduced in the paper.Chapter two:In this chapter, we give the criteria for some convexities of Orlicz spaces with Luxemburg norm, such as, compact uniformly conve -xity; weakly compact uniformly convexity; compact locally uniformly con -vexity; weakly compact locally uniformly convexity and k-drop convexity. And by it the equivalence of convexities in the Orlicz spaces are obained.Chapter three:In this paper, we give the criteria for some convexities of Orlicz spaces with Orlicz norm, such as, k-very convex space, k-uniform -ly extremely convex space, compact (weakly compact) uniformly convexi -ty, compact (weakly compact) locally uniformly convexity. And by it the equivalence of convexities in the Orlicz space are obained.Chapter four:In this chaper, we give the criteria for some smoothness of Orlicz spaces with Orlicz norm or Luxemburg norm(for example, k-unif -ormly smooth, k-very smooth, k-uniformly extremely smooth,etc ). And by the equivalence of smoothness in the Orlicz spaces are obained.