Some Geometric Properties Of Orlicz-Lorentz Spaces Equipped With The Orlicz Norm

In this paper,we explore some geometric properties of Orlicz-Lorentz spaces with Orlicz norm.It includes the uniform Kadec-Klee property,(compactly)local uniform convexity(k-)uniform convexity,midpoint local uniform convexity,and the characterization of extreme points and strong extreme points.In the first chapter,we give the description of the development of Orlicz-Lorentz space,the research background and status quo of geometric properties of this space are given collectively.As well as the main research contents of this paper in this.At the same time,some basic structures of this space are presented,together with some basic conclusions of this space.In chapter 2,the necessary and sufficient conditions for Orlicz-Lorentz sequence spaces equipped the Orlicz norms generated by general Orlicz function to possess uniform Kadec-Klee properties will be given in this chapter.According to the differences Orlicz norm expressions,the proof process will be divided into several cases.This method can also provide a new way to explore the Orlicz-Lorentz space generated by general Orlicz functions.In chapter 3,we give the necessary and sufficient conditions for the Orlicz-Lorentz function space equipped with Orlicz norm to process compactly local uniform convexity and local uniform convexity.The difficulty in proof process is that the dual space of the space has not been fully characterized yet.In this chapter,we give a new expression of Luxemburg norm,through which we can solve the problem under the condition that its dual space is still unknown.When we solve the necessity of strict convexity of functions,we give an important counterexample.At the same time,when it comes to the strict convexity of Orlicz functions,this counterexample proved to be a very useful tool in solving this such problems.In chapter 4,the necessary and sufficient conditions for Orlicz-Lorentz function spaces with Orlicz norms to have k-uniform convexity and uniform convexity are given in this chapter.In chapter 5,we characterize the strongly extreme points of Orlicz-Lorentz function spaces with Orlicz norms generated by N functions.Just like the chapter above,the necessary and sufficient conditions for the spaces to process the midpoint local uniform convexity are given.At the same time,we give the expressions of Orlicz norm and characterize the extreme points in Orlicz-Lorentz functions spaces generated by general Orlicz functions.