A note on ball-covering property of Banach spaces

作者：Cheng Lixin^*; Kadets Vladimir; Wang Bo; Zhang Wen

来源：Journal of Mathematical Analysis and Applications, 2010, 371(1): 249-253.

DOI：10.1016/j.jmaa.2010.04.076

• Ball-covering
• w*-Separability
• Banach space

By a ball-covering B of a Banach space X, we mean that B is a collection of open (or closed) balls off the origin whose union contains the unit sphere S(X) of X; and X is said to have the ball-covering property (BCP) provided it admits a ball-covering by countably many balls. In this note we give a natural example showing that the ball-covering property of a Banach space is not inherited by its subspaces: and we present a sharp quantitative version of the recent Fonf and Zanco renorming result saying that if the dual X* of X is w* separable, then for every epsilon > 0 there exist a (1 + epsilon)-equivalent norm on X, and an R > 0 such that in this new norm S(X) admits a ball-covering by countably many balls of radius R. Namely, we show that R = R(epsilon) can be taken arbitrarily close to (1+epsilon)/epsilon, and that for X = l(1)[0, 1] the corresponding R cannot be equal to 1/epsilon. This gives the sharp order of magnitude for R(epsilon) as epsilon -> 0.

• 出版日期2010-11-1
• 单位厦门大学