弱紧集与可逼近集的和

On the Sum of Weakly Compact Sets and Proximinal Sets

设C是Banach空间X的弱紧凸集,D是X的可逼近凸集(相应地,逼近弱紧凸集).利用弱紧凸集中序列的收敛性,证明了C+D也是可逼近集(相应地,逼近弱紧集),这是自反子空间与可逼近子空间的和(满足其和是闭的)仍然是可逼近子空间这一经典结论的推广和局部化.

Let C be a weakly comapct convex subset of a Banach space X and D be a proximinal convex subset （respectively,approximatively weakly compact convex subset） of X.By using the convergence of the sequences in weakly compact convex sets,it is proved that C＋D is also a proximinal set（respectively,approximatively weakly compact set）.This generalizes the classical result that the sum of reflexive subspace and proximinal subspace（satisfying the sum is closed） is again proximinal.