On K Convexity Of ψ-Direct Sums Of Banach Spaces

In this thesis,we mainly discuss the K convexity ofφ-direct sums of Banach spaces.There are four chapters in the paper.In the Introduction,we starts from the preliminary and some basic theorems and results.Also we introduce some conceptions and notations which will appear in the following chapters.In Chapter l,we use theφ-direct sums of (X (?) ... (?)X)φto characterize the geometrical properties of Banach spaces X.We give the characterizations of K strict convexity,K uniform convexity and uniform non-lN1-ness.In Chapter 2,we show that if X is K strictly convex and Y is L strictly con-vex,then X (?)φY is (K+L - l) strictly convex,whereφ∈Ψ2 is strictly convex,and generalize the result to the case ofφ-direct sums of finite Banach spaces.In addi-tion,we prove that X (?)φY has Banach-Saks property if and only if X and Y are Banach spaces with Banach-Saks property.In Chapter 3,we prove that if X is K uniformly convex and Y is L uniformly convex,then X (?)φY is (K+L-l) uniformly convex, whereφ∈Ψ2 is strictly con-vex,and generalize the result to the case ofφ-direct sums of finite Banach spaces.In addition,we discuss the K uniform convexity of X (?)1 Y,which is a special case.Last,we make a conclusion of our work,and point out some problems needed to be solved.