Some Properties Of Generalized Orlicz Spaces

According to the requirement of various theories and application, there are many generalizations of Orlicz spaces. The spaces equipped with the p-Amemiya norm are the generation of the Orlicz spaces. Extreme points, strongly extreme points and nonsquareness are important geometric properties in Orlicz spaces, and they have close connections with convexity, flatness and reflexity.In this thesis, some properties about the dual spaces of the Orlicz spaces and the strongly extreme points in the unit sphere and nonsquareness, uniformly nonsquareness and locally uniform nonsquareness of Orlicz spaces equipped with the p-Amemiya norms are studied. Contents of this thesis are divided into the following parts.First of all, some properties about the dual spaces of the Orlicz spaces are discussed. As a class of concrete Banach spaces, Orlicz spaces almost cover all classial Banach spaces, and their dual spaces are also a class of important Banach spaces. To study the properties of dual spaces of Orlicz spaces, we can understand the Orlicz spaces better, and we get some properties which are hold in the Orlicz spaces can be extended to its dual spaces.Second, the criteria for strongly extreme points in Orlicz spaces equipped with the p-Amemiya norm is studied. As well known, the strongly extreme points in Orlicz spaces equipped with the Luxemburg norm and Orlicz norm are very important pointwise properties. p-Amemiya norm is the generation of Luxemburg norm and Orlicz norm in Orlicz spaces, and in this section we give the criteria for Orlicz spaces equipped with the p-Amemiya norm.In the end, nonsquareness, uniformly nonsquareness and locally uniform nonsquareness of Orlicz spaces equipped with the p-Amemiya norm are studied. About the nonsquareness of Orlicz spaces equipped with the Luxemburg norm and Orlicz norm, many scholars studied a lot about them and got a lot of good results. In this section, we obtain that the Orlicz spaces equipped with the p-Amemiya norm are nonsquareness and locally uniform nonsquareness and give the sufficient conditions for uniform nonsquareness.