Some Pointwise Properties In Musielak-Orlicz Sequence Spaces

According to the requirement of various theories and application, there are many generalizations of Orlicz space and Musielak-Orlicz space is a common generalization. Point wise geometric property attenuate geometric property of global space and studying works from macro-property into micro-property is a qualitative leap in development of Orlicz spaces geometric theory. In this thesis, we mainly investigate some point wise properties of Musielak-Orlicz sequence spaces. The paper consists of five parts. The main results of this paper are summarized as following:First, it is reviewed that a survey of the theory of Orlicz spaces and Musielak–Orlicz spaces. Moreover, it is evaluating and summarizing the pioneer's main research results, and showing the background and significance of the content of each part in this thesis.Extreme point and strong-U point are both point wise property of rotundity. There are relations as well as differences between them. In chapter 2, we give the criteria of extreme point and strong-U point in Musielak-Orlicz sequence space. Meanwhile we get necessity and sufficient condition for that Musielak-Orlicz sequence space are rotundity as corollaries. In chapter 3, we present the criteria of H-point in Musielak-Orlicz sequence space. Furthermore, the sufficient and necessary condition for H-property is obtained.In 1975, Panda and Kapoor introduced the S property while working on the generalization of Frechet differentials. Afterwards, the concept of S point was introduced by Hao Cuixia est and the criteria in classical Orlicz spaces were obtained. In this paper, we give a criterion for S point in Musielak-Orlicz sequence space. Basing on that criterion, we provide a sufficient and necessary condition for S property for those spaces. At last, author present a criterion for compactly locally uniformly rotundity (CLUR) Point and weakly compactly locally uniformly rotundity (WCLUR) Point of Musielak-Orlicz sequence spaces and by it we get the criterion of locally uniformly rotundity (LUR) Point and global compactly locally uniformly rotundity (CLUR) property for those spaces.