Existence Of Nearest Points In Banach Spaces
Existence Of Nearest Points In Banach Spaces
复制标题
DOI：
10.4153/cjm-1989-032-7
复制
发表时间：
1989-08
期刊：
影响因子：
--
通讯作者：
J. Borwein;S. Fitzpatrick
中科院分区：
文献类型：
--
作者：
J. Borwein;S. Fitzpatrick
本文对作者所知道的（真实的）Banach空间的闭子集的最近点的存在性作了统一的发展。我们的工作是通过有条不紊地使用subderivatives简单。据我们所知，第3节和第7节的结果尤其是新的。在第五节和第六节中，我们给出了自反Kadec空间的Lau-Konjagin最近点刻画的精确证明（定理5.11，定理6.6），并给出了一个实质性的推广（定理5.12）。主要的开放问题是：在每个自反空间的每个闭子集的边界上最近点稠密吗？在自反空间中的真闭集真的不能有任何最近点吗？在第7节中，我们证明了存在一些非Kadec自反空间，其中最近点在每个闭集的边界上是稠密的。
This paper makes a unified development of what the authors know about the existence of nearest points to closed subsets of (real) Banach spaces. Our work is made simpler by the methodical use of subderivatives. The results of Section 3 and Section 7 in particular are, to the best of our knowledge, new. In Section 5 and Section 6 we provide refined proofs of the Lau-Konjagin nearest point characterizations of reflexive Kadec spaces (Theorem 5.11, Theorem 6.6) and give a substantial extension (Theorem 5.12). The main open question is: are nearest points dense in the boundary of every closed subset of every reflexive space? Indeed can a proper closed set in a reflexive space fail to have any nearest points? In Section 7 we show that there are some non-Kadec reflexive spaces in which nearest points are dense in the boundary of every closed set.