泛函分析课程中有界算子理论的教学探讨

A Discussion on Teaching of Bounded Operators in Functional Analysis

泛函分析课程是一门极其重要的数学专业课程,兼有理论性强和高度的抽象性等特点,采用传统的教师为主体的教学模式往往教学效果不佳.以有界算子理论这部分内容为例探索了研讨型教学模式,主要对比了l^1空间上有界线性算子全体与列和有界的无限维矩阵空间的等距同构关系,以及c0空间上有界线性算子全体与行和有界且每列元素趋于0的无限维矩阵空间的等距同构关系.讨论了序列空间l^∞到l^∞中的有界线性算子全体与无限维矩阵空间的关系.证明了行和有界的无限维矩阵可诱导出一个l^∞到l^∞的有界线性算子,通过反例说明了此对应不构成l^∞上的有界线性算子与行和有界的无限维矩阵空间的同构映射,丰富了泛函分析的教学内容和方法.

The course of Functional Analysis is an important professional course, which is characterized by extremely abstract and theoretical. The traditional teaching mode, which takes the teacher as the main body in the class, leads to poor teaching effect. Thus, the seminar teaching mode is practiced in the teaching of theory of bounded linear operators. Based on the discussion of the relationship between infinite matrix spaces and bounded linear operators on sequence spaces l1 and c0 respectively, the relationship between infinite matrix spaces and bounded linear operators on l∞ is discussed. That every infinite matrix can lead to a bounded linear operator on l∞ is proved. However, the correspondence isn’t an isomorphism, which is explained by an counter-example. Thus, the teaching content and method about bounded linear operators in the Function Analysis course is enriched.