从距离空间完备性角度探讨实数完备性理论  

On the Theory of Real Number Completeness from the Perspective of Distance Space Completeness

作　　者：胡永模 张海 HU Yongmo;ZHANG Hai(School of Mathematics and Physics,Anqing Normal University,Anqing 246133,China)

机构地区：[1]安庆师范大学数理学院,安徽安庆246133

出　　处：《安庆师范大学学报（自然科学版）》2021年第2期91-93,共3页Journal of Anqing Normal University(Natural Science Edition)

基　　金：安徽省高等学校质量工程项目(2019jyxm0288,2020szsfkc0557);安徽省高校自然科学研究项目(KJ2020A0503);安徽省省级教学示范课(1558)。

摘　　要：关于分析理论中完备性的相关理论,一般比较熟悉的是实数集完备性定理,但"数学分析"课程中实数集完备性定理的证明比较复杂,学生不太容易理解。为此,从"泛函分析"课程中的距离空间完备性的角度来阐述实数集完备性定理,把实数集看作为一个特殊的距离空间,得到实数完备性理论实际上是距离空间完备性理论的特殊情况的结论,从而帮助学生对实数的完备性理论有更加深入的理解。The completeness theory in analysis theory is generally familiar with the completeness theorem of real number set,but it is not easy for students to understand the process of proving the completeness theorem in the course of mathematical analysis.This paper attempts to expound the completeness theorem of real number set from the perspective of the completeness of distance space in the course of Functional Analysis.The main purpose of this paper is to regard real number as a special distance space and explain that the completeness theory of real number is actually a special case of the completeness theory of distance space,so as to help students have a more in-depth understanding of the completeness theory of real number.

关 键 词：距离空间 完备性 紧集 列紧集 

分 类 号：O177.1[理学—数学]