赋$\beta$-范空间中的最佳逼近问题

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

中文关键词:  局部$\beta$-凸空间  赋$\beta$-范空间  (赋范)共轭锥  最佳逼近.

英文关键词:locally $\beta$-convex space  $\beta$-normed space  normed conjugate cone  the best approximation.

中文摘要:

      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.

英文摘要:

      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.