Stability of the ball-covering property
Volume 250 / 2020
Studia Mathematica 250 (2020), 19-34
MSC: Primary 46B20; Secondary 46B03.
DOI: 10.4064/sm180607-6-10
Published online: 29 July 2019
Abstract
A normed space $X$ is said to have the ball-covering property (BCP, for short) if its unit sphere can be covered by the union of countably many closed balls not containing the origin. Let $(\varOmega, \varSigma, \mu)$ be a separable measure space and $X$ be a normed space. We show that $L_p(\mu, X)$ $(1\leq p \lt \infty)$ has the BCP if and only if $X$ has the BCP. We also prove that if $\{X_k\}$ is a sequence of normed spaces, then ${\mathbf X}=(\sum\oplus X_k)_p$ has the BCP if and only each $X_k$ has the BCP, where $1\leq p\leq\infty$. However, it is shown that $L_{\infty}[0, 1]$ fails the BCP.
