Some Geometrical Property Of Orlicz Spaces And Quotient Spaces

Some geometrical properties of Orlicz spaces equipped with Orlicz norm, Luxemburg norm and generalized Orlicz norm introduced by the thesis are investigated in this thesis. Moreover, some geometrical properties of quotient spaces are also studied .The main results of the thesis are summarized as following:Chapter 1 Introduction: It is reviewed that developing course of the theory of Orlicz spaces and the pioneer's main research results during more than 60 years old. Moreover, it is summarizing the construction of quotient spaces and showing the background and significance of the content of each part in this thesis.Chapter 2 -extreme points and -strongly extreme points in Orlicz spaces:Extreme points and strongly extreme points are fundamental concepts in geometry of Banach space. -extreme points and -strongly extreme points are extension of extreme points and strongly extreme points, respectively. In this paper, Criteria of -extreme points and -strongly extreme points of Orlicz spaces equipped with the Orlicz norm and the Luxemburg norm are given. Moreover, we get the sufficient and necessary condition of in Orlicz spaces.Chapter 3 Extreme points and strongly extreme points in Orlicz spaces equipped with the generalized Orlicz norm: In this paper, the conceptions of the generalized Orlicz norm and the generalized Luxemburg norm are introduced, and the criteria of extreme points and strongly extreme points of Orlicz function spaces equipped with the generalized Orlicz norm are obtained. Moreover, criteria of space strictly convex and mid-point locally uniform convex are given.Chapter 4 -rotundity and nearly strict convexity of quotient spaces. It is proved that ifis a -rotund or nearly strict convex Banach space andis a closed and proximinal subspace of, then the quotient spaceis also -rotund or nearly strict convex. Moreover, it points out that ifis an arbitrary Orlicz function such that , thenis not proximal inand quotient spaces is not -rotund or nearly strict convex (even ifis -rotund or nearly strict convex).