K一致极凸空间与K一致极光滑空间

K-UNIFORMLY EXTREMELY CONVEX SPACE AND K-UNIFORMLY EXTREMELY SMOOTH SPACE

引进K一致极凸空间与K一致极光滑空间的概念,它们分别是一致极凸空间与一致极光滑空间的推广.证明了K一致极凸性与K一致极光滑性具有对偶性质,即X 为K一致极凸(K一致极光滑)的,当且仅当X为K一致极光滑(K一致极凸)的;给出了K一致极凸(K一致极光滑)空间的3个特征刻画;证明了K一致极凸(K一致极光滑)蕴涵(K+1)一致极凸((K+1)一致极光滑),但反过来不成立;引进K-(WM) 性质,并利用K一致极光滑给出了自反的局部K一致光滑空间的特征刻画;证明了X 为局部K一致光滑,当且仅当X为K一致极凸且具有K-(WM)性质;证明了严格凸(光滑)的K一致极凸(K一致极光滑)空间是极凸(极光滑)空间.

In this paper,the definitions of K-uniformly extreme convexity and K-uniformly extreme smoothness which are the generalization of uniformly extreme convexity and uniformly extreme smoothness respectively are introduced,and it is proved that K-uniformly extreme convexity and K-uniformly extreme smoothness are the dual notion,i.e.,X~* is K-uniformly extreme convex (resp.K-uniformly extreme smooth) if and only if X is K-uniformly extreme smooth(resp.K-uniformly extreme convex),three characterizations of K-uniformly extreme convex (resp.K-uniformly extreme smooth) space are given,K-uniformly extreme convex (resp.K-uniformly extreme smooth) space implies (K+1)-uniformly extreme convex (resp.(K+1)-uniformly extreme smooth) space are proved,but their converses are not necessarily true,one characterization of the reflexive K-uniformly smooth space by introducing the property (K-(WM)~*) and using the K-uniformly extreme smoothness are given,it is also proved that X~* is locally (K-uniformly) smooth space if and only if X is K-uniformly extreme convex space and has the property (K-(WM)),and that strictly convex (resp.smooth_) K-uniformly extreme convex (resp.K-uniformly (extreme) smooth) space is extreme convex (resp. extreme smooth) space.