The Lifting Results Of Some Geometric Properties Of Banach Space In Sequence Space And Substitution Space

In half a century,with the rapid development of the geometric theory of Banach space,convexity and smoothness were well studied as one of the important content of it.The roughness of norm of Banach spaces was first mentioned to study the problem of Banach space differentiability. Roughness is actually a poor smoothness, so it has has been studied also.Up to now, a lot of research results about convexity, smoothness, differentiability, roughness and convergence of general Banach spaces have been obtained.It is necessary to study some geometric properties including these ones in Banach spaces can be promoted or not in sequence space and substitution spaces since some geometric properties can be promoted, and another can not. In 1970s,the notions of roughness and strongly roughness were introduced, and it has been studied.But the research from roughness extended k-roughness hasn't happened for a long time.In 2012,yiderihu introduced the notions about k-Rough norm and k-strongly rough norm and promoted them to the Banach sequence space lp(X,).In 1977,J.R. Partington discussed a broader class of Banach space including lp(Xt),namely substitution space PxXn,as a important Banach space,It is obvious that a lot of geometric properties have great research value in substitution spaces PxXn,So the lifting problem of k-Roughness and k-strongly roughness in substitution space PxXn will start to feel natural.The first chapter §1.2, we will finish the work in this aspect in this paper.Although the notions of weakly exposed points and strongly extreme points were introduced and its properties have been studied, but the research about the lifting properties of weakly exposed points and strongly extreme points in vector-valued sequence spaces cesp(xk) or substitution spaces PxXn have not been seen yet. So it is very necessary to discuss the above geometric properties.In the first chapter §1.3 and the second chapter §2.2,we just began its research and practice on it in this paper.In 2002,baiguozhong introduced the concept about average weak local uniform convexity,the lifting problems of average weak local uniform convexity in vector-valued sequence spaces were studied by Baolaiyou later.In this paper,the first chapter §1.4 discuss the lifting properties of average weak local uniform convexity in substitution space PxXn.The two types of vector-valued sequence space Xp(E) and ss(E) are different from the vector-valued sequence space cesp(xk). In the third chapter, we will research the lifting properties of weak* locally bicompactness in vector-valued sequence space Xp(E) and ss(E) in this paper.