Some Geometric Properties Of Orlicz Spaces Equipped With S-norms

As an important research content of Banach spaces,Orlicz spaces are widely used in various disciplines.With the efforts of mathematics workers for nearly a century,theory of Orlicz spaces is becoming more and more perfect.Orlicz spaces equipped with s-norms is a generalization of Orlicz spaces.It includes classical Orlicz space and all kinds of generalized Orlicz space.Its theory and application are more general.The exploration of this kind of space can not only enrich Orlicz space theory,but also provide intuitive data for Banach space.In this paper,the necessary and sufficient condition for Kadec-Klee property of Orlicz sequnce space and Orlicz function space equipped with s-norms and ? property of Orlicz sequence space equipped with s-norms are discussed.The main work is summarized as follows:Firstly,the purpose and significance of studying Orlicz space was introduced briefly in this chapter,introduced the development process of Orlicz spaces and the outstanding achievements of Orlicz space theory,and showed the research contents and conclusions of each part of this text.Secondly,the Kadec-Klee property of Orlicz sequence spaces equipped with s-norms was studied.As we all know,Kadec-Klee property is an important property in the geometric properties of Banach space,which is closely related to the fixed point theory.Many mathematicians have deeply studied the Kadec-Klee property of Orlicz space equipped with Orlicz norm,Luxemburg norm and p-Amemiya norm,and obtained many excellent results.In this chapter,we gave three equivalent propositions that Orlicz sequence spaces equipped with s-norms have Kadec-Klee property.Thirdly,the ? property of Orlicz sequence space equipped with s-norms was discussed,Banach spaces with ? property have Kadec-Klee property and normal structure and then it have the fixed point property.In this chapter,we gave a calculation formula of s-norms,and used this formula to characterize the equivalent proposition that Orlicz sequence space equipped with s-norms has ? property.Finally,the Kadec-Klee property of Orlicz function space equipped with s-norms was discussed.Due to the complexity of Orlicz function space equipped with s-norms,it is more difficult to solve the problem that Orlicz function space equipped with s-norms has Kadec-Klee property under the condition of unclear dual space.In this chapter,a new method is found,that is,using the characterization of the extreme point of Orlicz function space equipped with s-norms to solve the characterization of the Kadec-Klee property of Orlicz function space equipped with s-norms.