Banach空间可分性及可分商问题

On Separability of Banach Space and Separable Quotient Problem

可分商问题:是否每一个无限维的Banach空间都有一个无限维的、可分的商空间?这是一个至今都没有完全解决的问题.结合对该问题已有的等价转换条件,在系统研究了Banach空间在范数拓扑,w*拓扑和w拓扑中的可分性质以及讨论了它们之间的相互关系的基础上,先后得到在一般的Banach空间和经典Banach空间中可分商问题得以肯定回答的充分条件.研究结果一方面充实了Banach空间在3种常用拓扑关于可分的理论内容,另一方面也为可分商问题的进一步解决提供了丰富的理论基础.

The classical problem which is called separable quotient problem： does every infinite-dimensional Banach space admit a quotient,has been open by now.Based on several informed equivalent formulations of this famous unsolved problem,this paper firstly studies the separability of a Banach space under the norm-topology,w＊-topology and w-topology.The relationship among them is also discussed.Furthermore,some sufficient conditions for an affirmative answer to this question on some general and classical Banach spaces are successively derived.The results of this paper enrich the separability theory of Banach spaces,and provide a theoretical basis for further studying the separable quotient problem.