Asplund 空间的一些几何特征

SOME GEOMETRICAL CHARATERIZATIONS OF ASPLUND SPACE

本文主要讨论Asplund空间的一些几何特征。设 X 为 Banach 空间,本文证明了下述等价:(1)X 是 Asplund 空间;(2)X~*的每个有界范闭子集包含它的ω~*闭凸包的一个端点;(3)X~*的每个有界范闭子集包含它的凸包的一个端点;(4)对 X~*的每个有界范闭子集 A,存在 x_o∈X/{0}和 x_o~*∈A,使得 x_o~*(x_o)=(?)x~*(x_o);(5)对 X~*的每个有界范闭子集 A,集{x∈X,■x_o~*∈A,使得 x_o~*(x)=sup x~*(x)}在 X

In the paper,we prove the following theorem:hteorem 2.3 For a Banach space X,the following statemcnts areequivalent:(1)X is a Asplund space;(2)Every closed bounded subset of X~* contains an extreme point ofits weak~* closed convex hull;(3)Every closed bounded subset of X~* contains an extreme point of itsconvex hull;(4)For each closed bounded subset A of X~*,there is a nonzero x_0∈Xand x_0~*∈A,such that x_0~*(x_0)=(?) x~*(x_0);(5)For each closed bounded subset A of X~*,the set {x∈X;(?) x_0~*∈Asuch that x_0~*(x)=(?)(x)} is norm dense in X.