Some Geometric Properties Of Banach Sequence Spaces

Banach space theory is an important research direction of functional analysis,and it is also a research subject with theoretical significance and application value incurrent mathematics. The establishment and development of it not only expands thefunctional analysis course content, but also has brought a broader application forother areas of science and technology. The geometric properties of Banach space arethe important research content in Banach space theory. In this paper, we study somegeometric properties about two kinds of the specific Banach sequence space—Cesarosequence space and Musielak-Orlicz sequence space. The full text is divided intothree parts, the main work is as follows:The first chapter is introduction. This chapter presents the purpose andsignificance of the research, and detail elaborated research development status athome and abroad of Banach space geometric theory, Finally this chapter shows themain content of research in this paper.In the second chapter, we firstly promote the definition of strongly rotundproperty and compactly strong rotund property in Banach space, introduce theconcept of strongly rotund point and compactly strong rotund point, and give therelationships with strongly rotund point and compactly strong rotund point in Banachspace. Then we discuss the characterization of compactly strong rotund property in aconcrete Banach space—Cesaro sequence spaceces p, and we show that the spaceces pposses property K and weak orthogonality when1＜p＜∞.In the third chapter, we study local β-property in Musielak-Orlicz sequencespace equipped with the Orlicz norm. First of all, we give the criteria of β-point inMusielak-Orlicz sequence space equipped with the Orlicz norm, thus get thesufficient and necessary condition of which the space has the local β-property.