Ball-covering property of Banach spaces that is not preserved under linear isomorphisms  

作　　者：CHENG LiXin CHENG QingJin LIU XiaoYan 

机构地区：[1]Department of Mathematics,Xiamen University,Xiamen 361005,China

出　　处：《Science China Mathematics》2008年第1期143-147,共5页中国科学：数学（英文版）

基　　金：Supported by the National Natural Science Foundation of China (Grant No. 10471114)

摘　　要：By a ball-covering B of a Banach space X, we mean that it is a collection of open balls off the origin whose union contains the sphere of the unit ball of X. The space X is said to have a ball-covering property, if it admits a ball-covering consisting of countably many balls. This paper, by constructing the equivalent norms on l~∞, shows that ball-covering property is not invariant under isomorphic mappings, though it is preserved under such mappings if X is a Gateaux differentiability space; presents that this property of X is not heritable by its closed subspaces; and the property is also not preserved under quotient mappings.By a ball-covering B of a Banach space X, we mean that it is a collection of open balls off the origin whose union contains the sphere of the unit ball of X. The space X is said to have a ball-covering property, if it admits a ball-covering consisting of countably many balls. This paper, by constructing the equivalent norms on l ∞, shows that ball-covering property is not invariant under isomorphic mappings, though it is preserved under such mappings if X is a Gateaux differentiability space; presents that this property of X is not heritable by its closed subspaces; and the property is also not preserved under quotient mappings.

关 键 词：BALL-COVERING isomorphic invariant Gateaux differentiability space Banach space 46B20 46G05 

分 类 号：O177.2[理学—数学]