本课程旨在为应用数学系的研究生提供现代分析的基本思路和工具。这包括真正的分析和功能分析的主题，在二十世纪和之后发展的分析。我们将在第二学期主要分析第一学期的实际分析和功能分析。希望能为其他一些理论或应用领域的进一步应用打下坚实的基础。在选修这门课程后，学生们将对现代分析问题的方法有大体的认识。
       This is the course designed to acquaint the graduate students in the applied mathematics department with basic ideas and tools in modern analysis. This comprises the subjects of real analysis and functional analysis, the analysis developed in the 20th century and after. We will treat real analysis mainly in the first semester and functional analysis in the second. Hopefully, we can lay down a solid foundation for further usage in some other theoretical or applied area. After taking the course, the students are expected to have the general idea on the modern ways to attack the analysis problems.


     課程目標/概述  

This is the course designed to acquaint the graduate students in the applied mathematics department with basic ideas and tools in modern analysis. This comprises the subjects of real analysis and functional analysis, the analysis developed in the 20th century and after. We will treat real analysis mainly in the first semester and functional analysis in the second. Hopefully, we can lay down a solid foundation for further usage in some other theoretical or applied area. After taking the course, the students are expected to have the general idea on the modern ways to attack the analysis problems.


課程章節  

章節

內容綱要
第一章
Measure theory

 
Abstract measure theory
Lebesgue measure
第二章
Integration

 
Convergence theorems (Monotone Convergence theorem,
Dominated Convergence theorem, Fatou lemma)
Fubini theorem
Tonelli theorem
Lpspace
Completeness
Compactness
第三章 Metric spaces
Arzela-Ascoli theorem

 

課程書目 

A. Friedman, Foundations of modern analysis, Dover, New York,