关于联合可逼近子空间和的一个注记

(华侨大学数学科学学院,福建 泉州 362021)

可逼近; 联合可逼近; 弱紧集

A Remark on the Sum of Simultaneously Proximinal Subspaces
MENG Qingfeng*,LUO Zhenghua,SHI Huihua

(School of Mathematical Sciences,Huaqiao University,Quanzhou 362021,China)

proximinal; simultaneously proximinal; weakly compact sets

DOI: 10.6043/j.issn.0438-0479.201611011

备注

设G是Banach空间X的闭子集.G称为在X中是联合可逼近的(simultaneously proximinal),如果对每个有界集AX,都存在g∈G,使得d(A,G)≡infu∈Gsupa∈A‖a-u‖=supa∈A‖a-g‖.证明了Banach空间中的弱紧凸集与联合可逼近凸集的和是联合可逼近的.作为推论,证明了对于Banach空间X的自反子空间F和联合可逼近子空间G,如果F+G 是闭的,则F+G是联合可逼近的.

Let G be a closed subset of a Banach space X. Then G is said to be simultaneously proximinal in X if, for every bounded set AX,there exists a g∈G such that d(A,G)≡infu∈G supa∈A ‖a-u‖=supa∈A ‖a-g‖.In this paper,we prove that,if C and D are two convex subsets of a Banach space X,where one is weakly compact and the other is simultaneously proximinal,then C+D is simultaneously proximinal.As a consequence,we prove that if F and G are respectively reflexive subspace and simultaneously proximinal subspace of a Banach spaces X such that F+G is closed,then F+G is simultaneously proximinal.