Some Properties Of Generalized Orlicz Spaces

According to the requirement of various theories and application, there are many generalizations of Orlicz spaces. The spaces LM ,p equipped with p—Amemiya norm are the generation of the Orlicz spaces. k—extreme point, k—strongly extreme point,λ—property and uniformlyλ—property are important geometric properties in Orlicz spaces, and they have close connections with k—strict convexity,mid—local k—uniformly convexity property,extreme point and strict convexity.In this paper, some properties,namely k—extreme point,k—strongly extreme point,λ—property and uniformlyλ—property are discussed in the spaces LM ,p. Contents of this thesis are divided into the following parts.First, the criterion of the k—extreme point and k—strict convexity property in LM ,p spaces. As well known, k—extreme point is important geometric propertie in theory of Banach spaces,p—Amemiya norms are the extension of Luxemburg norm and Orlicz norm,in this part,we get a necessary and sufficient condition that a point on the S ( L_{M ,p}) is a k—extreme point. Furthermore,we get the criteria for LM ,p spaces are k—strict convexity.Second, the problems of k—strongly extreme point and mid—local k—uniformly convexity property in Orlicz spaces equipped with p—Amemiya norm. In the spaces equipped with Luxemburg norm and Orlicz norm ,the criteria for k—strongly extreme point and mid—local k—uniformly convexity property have given out,in this part,we give the criteria that a point on the S ( L_{M ,p})is a k—strongly extreme point. From the result,we get the necessary and sufficient condition that LM ,p spaces are mid—local k—uniformly convexity.In the end,λ—property and uniformlyλ—property of Orlicz spaces equipped with p—Amemiya norm are studied. About theλ—property and uniformlyλ—property of Orlicz spaces equipped with Luxemburg norm and Orlicz norm,many scholars studied a lot about that,and got a lot of good results. In this section,we obtain that L_{M ,p} spaces have theλ—property. And obtain the necessary and sufficient condition that L_{M ,p} spaces have the uniformlyλ—property.