Extreme Points Of Orlicz-Lorentz Spaces

Since A.Kami(?)ska gave the notion of Orlicz-Lorentz space in 1990,many conclusions about the Orlicz-Lorentz space with the Luxemburg norm have been obtained. Wu and Ren's giving the Orlicz norm of Orlicz-Lorentz space in 1999,the results about the Orlicz-Lorentz space with this norm was sporadic,and lack of system.The main work of this paper is:1.Discussed norm theorems,convergence theories and subspacees theorems in Orlicz-Lorentz sequence space,which had not been checked. 2.Gave the the characterization of extreme points in Orlicz-Lorentz sequence space with the Luxemburg norm.3.Gave the the characterization of extreme points in function space with the Orlicz norm.We verified that:Ifω(t) > 0,then x∈S(∧_φ,ω⁰) is an extreme point if and only if for everyκ∈K(x),eitherκ|x(t)| = g(t)∈S,orκ|x(t)| =αχA,withα∈S′and m(σ(A)∩L(ω)) = 0.This paper has three chapters,each of them investigated different aspect respectively.The first chapter laid a fundation for this paper,it mainly discussed the corresponding fundamental theories in Orlicz-Lorentz sequence space which combined with the existing fundamental theories of Orlicz space,it also laid a fundation for the investigation of extreme points in the second and thrid chapters.The second chapter kept on to consider the sequence space.Referenced to the theory of extreme points in function space with the Luxemburg norm,obtained the characterization of extreme points in sequence space with the Luxemburg norm.In the last chapter,we turned to the function space.It is also the most difficult chapter.We mainly studied the characterization of extreme points in Orlicz-Lorentz space with the Orlicz norm.