Some Geometric Properties Of Banach Spaces And Orlicz Spaces

This dissertation mainly focuses on some geometric properties of Banach spaces and the particular Banach spaces-Orlicz spaces. The dissertation consists of six chapters and the main results of this thesis are summarized as follows:In Chapter one, we introduce the developing history and background of the theo-ries of Banach spaces and Orlicz spaces, and show the main contents of this thesis.In Chapter two we study the k-dentability of Banach spaces. As a generalization of the notion of dentability, we introduce some definitions of k-dentability in Banach spaces. Also, we discuss the properties of various k-dentability, and describe the char-acterization of Banach spaces that have certain k-convexity by using the above no-tions. In addition, we also shows that k-dentability implies (k+1)-dentability, but its converse is not necessarily true.In Chapter three we consider compactly strongly convex property and weakly compactly strongly convex property of Banach spaces. We first introduce the concep-tions of compactly strong convexity and weakly compactly strong convexity in Ba-nach spaces, they are the generalizations of the notions of strong convexity and weakly strong convexity. Then we investigate the relationships between compactly strongly convex and weakly compactly strongly convex properties and some other geometric properties, and prove that the compactly strongly convex property is the dual property of S property and the weakly compactly strongly convex property is dual to WS prop-erty. Finally, in a special kind of Banach space-Orlicz sequence spaces equipped with the Orlicz norm lMO, we present the concretely depiction of compactly strongly convex property and weakly compactly strongly convex property.In Chapter four we discuss the properties of Hl and Hg in Orlicz spaces equipped with the p-Amemiya norm. We prove that in the case of a nonatomic infinite measure space, properties Hl and Hg for (LM,‖·‖M,p) do not coincide. More specifically, we prove that if the generating function M vanishes only at zero, then properties Hl and Hg coincide and they are equivalent to M∈△2 and M is finitely valued; if M vanishes outside zero, then properties Hl and Hg differ. Analogous results are also proved for the subspace EM of order continuous elements of the space LM·In Chapter five we investigate the S property of Musielak-Orlicz sequence spaces. In this chapter, we give the criteria for S points in Musielak-Orlicz sequence spaces and basing on those criteria, we provide the necessary and sufficient conditions for S property for those spaces, improving and extending the discussion about S property.In Chapter six we are concerned with several kinds of pointwise properties in Musielak-Orlicz sequence spaces equipped with the Orlicz norm. Extreme points,str-ong U points,locally uniformly rotund points and weakly locally uniformly rotund points are basic notions in the geometry of Banach spaces and they have extensive application. In classical Orlicz spaces, the criteria for these pointwise properties have been obtained. However, because of the complicated structure of the Musielak-Orlicz sequence spaces, criteria for those pointwise properties mentioned above have not been found in this class of spaces. In the first section of this chapter, we present the criteria for extreme points and strong U points in Musielak-Orlicz sequence spaces equipped with the Orlicz norm. Meanwhile we get the necessary and sufficient condition for that these spaces mentioned above are rotund as corollaries, and we also give a example to illustrate that strong U points are essentially stronger than extreme points in these spaces. In the second section, we give a criterion for locally uniformly rotund points and weakly locally uniformly rotund points, in consequence to get the equivalent con-dition that those spaces are locally uniformly rotund and weakly locally uniformly rotund, thereby we solve the problem which has not been solved before.