范数可微性和Banach空间的一致球覆盖性质

Differentiability of Norm and Uniform Ball-Covering Property in Banach Space

在这篇文章中,作者首先给出了范数在集合上一致光滑的定义,而且证明了存在一个l^(∞)的一致球覆盖,使得l^(∞)的范数在球覆盖点是一致光滑的.其次,作者证明了如果(П_(i=1)^(2)X_(i),‖·‖_(p))是一个乘积空间,这里p∈[1,+∞],则存在(П_(i=1)^(2)X_(i),‖·‖_(p))的一个一致球覆盖,使得(П_(i=1)^(2)X_(i),‖·‖_(p))的范数在球覆盖点是一致光滑的当且仅当存在X_(i)的一个一致球覆盖,使得X_(i)的范数在球覆盖点是一致光滑的.最后,作者证明了如果X是一致光滑空间且可分,则存在两个序列{x_(n)}_(n=1)^∞■X和{r_(n)}_(n=1)^(∞)■R,使得:(1)存在{x_(n)}_(n=1)^(∞)的一个子序列{X_(j)}_(j=1)^(∞),使得{‖x_(j)‖^(-1)x_(j)}_(j=1)^(∞)上的每一点都是B(X)的强暴露点;(2)对每个n∈N,‖x_(n)‖^(-1)x_(n)是B(X)的端点;(3)集序列{B(x_(n),r_(n))}_(n=1)^(∞)是X的一个一致球覆盖.

In this paper,the author first gives the definition which norm uniformly smooth on a set,and proves that there exists a uniformly ball-covering of l^(∞) such that the norm of l^(∞) is uniformly smooth on ball-covering points.Secondly,the author proves that if (П_(i=1)^(2)X_(i),‖·‖_(p))is a product space,where p∈[1,+∞],then there exists a uniformly ball-=1 covering of(П_(i=1)^(2)X_(i),‖·‖_(p))such that the norm of(П_(i=1)^(2)X_(i),‖·‖_(p))is uniformly smooth on ball-covering points if and only if there exists a uniformly ball-covering of X_(i)such that the norm of X_(i)is uniformly smooth on ball-covering points.Finally,it is proved that X is a uniformly smooth space and separable,then there exist two sequences{x_(n)}_(n=1)^∞■X and{r_(n)}_(n=1)^(∞)■R such that(1)There exists a subsequence{x_(n)}_(n=1)^(∞) such that{‖x_(j)‖^(-1)x_(j)}_(j=1)^(∞) is a sequence of strongly exposed points of B(X);(2)For each n∈N,the point ‖x_(n)‖^(-1)x_(n)is an extreme point of B(X);(3)The set sequence{B(x_(n),r_(n))}_(n=1)^(∞)is a uniformly ball-covering of X.