Banach空间中的完备集
2017-06-10吴森林张新玲计东海
吴森林+张新玲+计东海
摘要:针对Banach空间中完备集的相关问题, 回顾了完备集这一概念的来源:等宽集的一些基本性质, 介绍了完备集的一些性质以及与完备集相关的若干研究问题和相关结果。 结果表明, 围绕Banach空间中的完备集及其相关问题还有很多待完成的工作。
关键词:Banach空间; 等宽集; 完备集; 完备化集
DOI:10.15938/j.jhust.2017.02.016
中图分类号: O177
文献标志码: A
文章编号: 1007-2683(2017)02-0083-05
Abstract:For the related problems of complete sets in Banach spaces, some fundamental properties of sets of constant width which is the origin of the concept of complete sets are reviewed, and properties of complete sets and research problems and corresponding results related to complete sets are also presented. It is shown that there are much research to be done concerning complete sets and related problems in Banach spaces.
Keywords:Banach spaces; sets of constant width; complete sets; completion of sets
6完備化集与其他特殊凸集类的关系
设A是有限维Banach空间中的一个凸体, 若任何一个真包含于A的凸体的最小宽度均严格小于A的最小宽度(A的平行的支撑超平面之间距离的下确界), 该凸体称为不可缩的(reduced)。显然的, 任意一个等宽集都是不可缩的。文[46]中声称有限维Banach空间中任何一个完备集均是不可缩的, 然而, Martini和吴森林已经给出一个反例说明该结论是不正确的(参见文[45])。因此, 在有限维Banach空间乃至无穷维Banach空间中考虑不可缩凸集与完备集的关系十分有必要。关于
瘙 綆 n和有限维Banach空间中不可缩凸体的更多内容请参见文[46]和[47]以及这两篇综述文章中所列文献。
7结语
尽管很多数学家在一般的实Banach空间特别是有限维实Banach空间中围绕着完备集及其相关性质, 集合的完备化映射以及与完备集有关的若干问题已经做了一系列重要的工作, 但是关于完备集仍然有很多未解决的问题, 希望本文对完备集相关问题的介绍能让更多的人关注并尝试解决这些问题。
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(編辑:温泽宇)