Strongly Convex Points And Property H_? Of Orlicz Spaces

In this paper,we mainly focus on some pointwise properties and some geometric properties of Banach spaces and Orlicz spaces.This paper consists of three chapters and the main results of this paper are summarized as follows:In Chapter one,we introduce the development history and the background of the theories of Banach spaces and Orlicz spaces,decribe the singnificance of research pointwise properties and geometric propertries in Banach spaces,and give the main contents of this paper.In Chapter two,we consider strongly convex point in Orlicz spaces.We firstly make some geometric properties of Banach spaces be pointwise,introduce the concept of weak strongly convex point and weak compactly strongly convex point.We discuss the relationships between them and strongly convex point,compactly strongly convex point,locally uniformly rotund point,strongly smooth point,H point,WM point,strongly U point,S point and so on.Then we study strongly convex points in a special kind of Banach space—Orlicz spaces.According to the relationship between strongly convex point and stongly extreme point,we present a criterion for strongly convex point in sequence spaces and functions spaces of Orlicz equipped with Luxemburg norm.As a corollary we provide the necessary and sufficient conditions for strongly convex for those spaces.In Chapter three,we consider property H_? for Orlicz function spaces equipped with the p-Amemiya norm.We discuss the relationship between the property H for convergence in measure and the?₂-condition for Orlicz function spaces L_M,P equipped with the p-Amemiya norm.From our results it follows that if Orlicz Function M vanishes only at zero,then the Orlicz function spaces equipped with the p-Amemiya norm L_M,P have property H_? if and only if Orlicz functionM satisfies the?₂-condition for all real numbers and M is finitely valued;if M vanishes outside zero,then L_M,P have property H_? if and only if M satisfies the?₂-condition at infinity and M is finitely valued.Analogous results are also proved for the subspace E_M,P of the Orlicz function space L_M,P.