Compactly Strongly Convex Property And K-Property Of Orlicz Spaces

Orlicz spaces is a kind of concrete Banach spaces, which plays a very important role in doing research work on theory of Banach spaces and applications. It not only provides methods of theory research and a lot of counter-examples for the general Banach spaces, but also provides space frame for the application of Banach spaces.Therefore, the study of Orlicz spaces has important theoretical significance and practical value. In this paper, we mainly discuss two kinds of geometric properties of Orlicz spaces. The paper consists of three parts, the main content is as follows:In the first chapter, the origin, purpose and significance of the study of this dissertation are introduced. The theory research situation of Orlicz spaces and Musielak-Orlicz spaces at home and abroad are expounded, and the main contexts of this dissertation are given.In the second chapter, we investigated the depiction for compactly strongly convex property in Orlicz function spaces L_M~0 equipped with the Orlicz norm by the dual relationship of convexity and smoothness in the geometric theory of Banach spaces. Firstly, we gave the criterion for S property in the subspace E_M of Orlicz function spaces L_E equipped with the Luxemburg norm. Secondly, we presented the specific characterization for compactly strongly convex property in Orlicz function spaces L_M~0 equipped with the Orlicz norm, and then the necessary and sufficient condition that these spaces have strongly convex property was obtained.In the last chapter, the k-property is an important geometric property in Banach spaces, the k-property is closely relately to the weak formal structure and the weak fixed point property. In this chapter, we investigated the problem how to characterize the k-property in the Musielak-Orlicz sequence spaces, the necessary and sufficient conditions for the k-property of the Musielak-Orlicz sequence spaces equipped with Orlicz norm and Luxemburg norm is gived, therefore, we got a sufficient condition that the Musielak-Orlicz sequence spaces has the weak fixed point property.