度量空间的非标准特性（Ⅱ）

The Nonstandard Character of Metric Spaces(Ⅱ)

本文在文《度重空间的非标准特性（Ⅰ）》的基础上，用非标准分析方法刻划了度量空间上的全有界映射与紧映射的许多特性，并给出了一个赋范线性空间的维数有限的非标准特征。最后，在赋范线性空间中，通过弱拓扑性质的非标准刻划，简洁地证明了Ｅｂｅｒｌｅｉｎ－Ⅲ定理。

In the paper, we continue the research of [1].With the nonstandard analytical method,many properties of the totally bounded mapping and the compact mapping are described in the metric space.The nonstandard character that a norm space is a finite dimension space is given. Finally,by the nonstandard description of the weak topological properties,Eberlein theorem is succinctly proved in the norm space.