Some Properties Of Convexity And Smoothness Of Orlicz-Bochner Sequence Spaces

In the 30's of the 20th century,in order to solve some problems of the nonlinear analysis based on the space Lp,the Polish mathematician Orlicz generalized the space Lp.He generalized the function M(u)=|u|p(P>1)into the N function,that is,he generalized the space Lp into the Orlicz space.Some nonlinear analysis problems was effectively figured out due to the nature of the Orlicz space.Thus it is interesting for people to study its geometric properties.Orlicz-Bochner space is a kind of generalization of Orlicz space.It has a more complicated structure than that of the Orlicz space.Hence,the theory of geometry of the Orlicz-Bochner space still has a long way to go.In this dissertation,some kind of geometric properties of the Orlicz-Bochner spaces are studied.The structure of this dissertation is listed as followsIn the first chapter,we introduce the research background,history and statusIn the second chapter,some definitions,theorems and lemmas of the Banach spaces,the Orlicz sequence spaces and the Orlicz-Bochner sequence spaces are re-calledIn the third chapter,the criteria for the uniformly non-square point of the Orlicz-Bochner sequence spaces endowed with the Orlicz norm are givenIn the fourth chapter,the criteria for the strong and very smooth points of Orlicz-Bochner sequence spaces equipped the Orlicz and the Luxemburg norm are givenIn the fifth chapter,the necessary and sufficient conditions for the weak*uni-form rotundity of the Orlicz-Bochner sequence spaces endowed with the Orlicz norm are given.