赋$\beta$-范空间中的最佳逼近问题

本文讨论赋$\beta$-范空间中的最佳逼近问题.以1]引进的共轭锥为工具,借助2]中关于$\beta$-次半范的Hahn-Banach延拓定理,第二节给出赋$\beta$-范空间的闭子空间中最佳逼近元的特征,第三节得到赋$\beta$-范空间中任何凸子集或子空间均为半Chebyshev集的充要条件是空间本身严格凸,文章最后证明了严格凸的赋$\beta$-范空间中任何有限维子空间都是Chebyshev集.

The Problems of Best Approximation in $\beta$-Normed Spaces~($0<\beta<1$)

This paper deals with the problems of best approximation in $\beta$-normed spaces. With the tool of conjugate cone introduced in 1] and via the Hahn-Banach extension theorem of $\beta$-subseminorm in 2], the characteristics that an element in a closed subspace is the best approximation are given in Section 2. It is obtained in Section 3 that all convex sets or subspaces of a $\beta$-normed space are semi-Chebyshev if and only if the space is itself strictly convex. The fact that every finite dimensional subspace of a strictly convex $\beta$-normed space must be Chebyshev is proved at last.