关于Eberlein-mulian定理的注记

A note on the Eberlein-mulian theorem

基于对证明方法的分析,利用度量化技巧与Whitley构造技术,给出Eber-lein-mulian定理的一个细致简单的证明.此证明不借助完备性,因此不影响定理的实质;仅用到Banach-Alaoglu定理与Hahn-Banach定理,因此容易被初学者理解.

The Ebcrlein-Smulian theorem is one of the basic and profound results in functional analysis. There are many proofs of it. In this paper, based on the analysis of the proofs, a brief and careful proof of the Eberlein-Smulian theorem is given by use of the metrization skill and the Whitley structure. The completeness is not necessary in the proof, thus it will not hurt the essence of the theorem at all. Only the Banach-Alaoglu theorem and the Hahn-Banach theorem are used in the proof, so as to be easy for students to understand.