Stability of the ball-covering property
作者: Zhenghua LuoBentuo Zheng
作者单位: 1School of Mathematical Sciences Huaqiao University Quanzhou, Fujian, China, 362021
2Department of Mathematical Sciences University of Memphis Memphis, TN 38120, U.S.A.
刊名: Studia Mathematica, 2020, Vol.250 , pp.19-34
来源数据库: Institute of Mathematics Polish Academy of Sciences
DOI: 10.4064/sm180607-6-10
原始语种摘要: 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 countablymany closed balls not containing the origin. Let $(\varOmega, \varSigma, \mu)$ bea 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 $X$ hasthe BCP. We also prove that if $\{X_k\}$ is a sequence of normedspaces, then ${\mathbf X}=(\sum\oplus X_k)_p$ has the BCP if andonly each $X_k$ has the BCP, where $1\leq p\leq\infty$. However, it isshown that $L_{\infty}[0, 1]$ fails the BCP.
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  • covering 覆盖物
  • property 所有权
  • separable 可分的
  • where 哪里
  • sphere 
  • covered 有盖的
  • normed 赋范的
  • short 短的
  • measure 测度
  • sequence 次序