| As a very important branch of modern functional analysis,the geometric theory of Banach space has great significance and research value in the study of modern mathematics.Orlicz space is a special Banach space.Owing to the variety of generating functions,Orlicz space has has various forms and different personalities.This feature enables it to provide sufficient examples and counterexamples for more abstract Banach spaces,and the techniques and methods of describing Orlicz spaces in their geometric properties can provide reference ideas for solving geometric problems in more general spaces.Because of the important theoretical properties and application value of Orlicz space,many mathematicians regard it as an important research direction.So far,the research on Orlicz space theory with Orlicz norm and Luxemburg norm has been relatively mature.In this paper,the geometric properties and applications of Orlicz spaces with p-Amemiya norm and s-norm are studied.The results can provide a new theoretical support for the study of geometric properties of Banach spaces.The full thesis is divided into four parts,as follows:Firstly,this gives a brief review of the developmen of Orlicz space,and Orlicz space theory in domestic and foreign research present situation has carried on the brief narration.Secondly,we introduce a new geometric constant called nearly uniform smoothness modulus.Then the direct formula for the nearly uniform smoothness modulus in the Orlicz sequence space equipped with p-Amemiya norm is given.Finally,the sufficient and necessary conditions for nearly uniformly smooth and have the property of fixed point of the Orlicz sequence space equipped with p-Amemiya norm are obtained.Then,we study the extreme point and the strictly convex of a generalized Orlicz space—Orlicz spaces equipped with s-norm.Firstly,some basic properties of s-norm are discussed.Then,the criterion of extreme points of Orlicz space equipped withs-norm is obtained and the sufficient and necessary conditions for strictly convex of the generalized Orlicz space are given.Finally,on the basis of the properties of s-norms obtained in Chapter 3,we study strongly extreme points and middle point locally uniformly convex in Orlicz spaces generated by Orlicz function and equipped with s-norm.Finally,using a new technique,the criterion for strongly extreme points is given and the sufficient and necessary conditions for middle point locally uniformly convex are obtained. |
