张量积空间的最佳逼近

BEST APPROXIMATIONS IN TENSOR PRODUCTS

设X是Banach空间,E为X的闭子空间,若对任意的x∈X,一定存在y∈E使得 d(x,E)=||x-y||,则称E在X中有最佳逼近性质.讨论给定Banack空间的闭子空间的最佳逼近性质是逼近论中的一个重要问题.本文在张量积空间中讨论了这类问题. 设X为一个Banach空间,E是X的闭子空间.如果存在闭子空间E′使X=EE′,且对于任意的x=g+g′∈X,g∈E,g′∈E′,有||x||=||g||+||g′||。

In this paper we study some M-ideal density problems closely related to the bestapproximations in tensor products. We prove that if X and Y are two Banach spacessuch that (X(×)Y) = (X(×)Y), G is an M-ideal of X and E is a subspace of X.Assume that G ∩ E is G-dense in E. Then (G ∩ E)(×)Y is (G(×)Y)-dense inE(×)Y. A similar result is obtained for tensor products of C-algebras.