Bochner-Lebesgue空间内的最佳同时逼近

在本文中,我们研究了Bochner-Lebesgue空间内的相对于欧氏空间的Minkowski范数的最佳同时逼近.首先,给出了由距离函数表示的最佳同时逼近的刻画.然后,利用可测选择定理证明其函数取值于一个闭的可分子空间的Bochner-Lebesgue空间,其同时可逼近性等价于此闭的可分子空间的同时可逼近性.最后,指出子空间的可分性是同时可逼近性等价的必要条件.

Best Simultaneous Approximations in Bochner-Lebesgue Spaces

In this paper, we consider the best simultaneous approximations in Bochner-Lebesgue spaces with respective to Minkowski'norms in Euclidean spaces. Firstly, we give a characterization of best simultaneous approximations by the distance func-tions. Then,by applying this characterization and a measurable selection theorem we show the simultaneous proximinality of a Bochner-Lebesgue space whose func-tions take values in a closed separable subspace is equivalent to the simultaneous proximinality of the closed separable subspace. Finally, we conclude that for their equivalence,the separability of the subspace is necessary.