Real Analysis I
本課程是由 國立陽明交通大學應用數學系 提供。
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.
課程用書：A. Friedman, Foundations of modern analysis, Dover, New York, 1982.
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| Instructor(s) | Department of Applied Mathematics Prof. Pei Yuan Wu |
|---|---|
| Course Credits | 3 Credits |
| Academic Year | 99 Academic Year |
| Level | Master Student |
| Prior Knowledge | Advanced calculus and general mathematical maturity. |
| Related Resources | Course Video Course Syllabus Course Calendar |
| Week | Course Content | Course Video |
|---|---|---|
| Class1 課程介紹 第一章 Measure theory Class1 課程導論 | Watch Online | |
| Class2 | 線上觀看 | |
| Class3 Sec.1.2. Measure Sec.1.3. Outer Me | 線上觀看 | |
| Class4 Sec.1.4. Constructing outer measure | 線上觀看 | |
| Class5 Sec.1.5-1.6 Lebesgue measure | 線上觀看 | |
| Class6 Sec.1.7 Metric space | 線上觀看 | |
| Class7 Sec.1.8-1.9 Construction of metric outer meas | 線上觀看 | |
| Class8 Sec.1.9 Construction of metric outer measure | 線上觀看 | |
| Class9 sec.1.10 Signed measure | 線上觀看 | |
| Class10 | 線上觀看 | |
| 第二章 Integration Class11 Sec. 2.2 Operations on measurable functions | 線上觀看 | |
| Class12 Sec 2.3. Egoroffs Thm. | 線上觀看 | |
| Class13 Sec 2.3 Egoroffs Thm. | 線上觀看 | |
| Class14 Sec 2.4 Convergence in measure | 線上觀看 | |
| Class15 Sec 2.5 Integrals of simple functions | 線上觀看 | |
| Class16 Sec. 2.6 Integrable functions | 線上觀看 | |
| Class17 | 線上觀看 | |
| Class18 Sec. 2.7 Properties of integral | 線上觀看 | |
| Class19 | 線上觀看 | |
| Class20 | 線上觀看 | |
| Class21 Sec.2.9 DCT | 線上觀看 | |
| Class22 Sec. 2.10 Applications of DCT | 線上觀看 | |
| Class23 Sec 2.11 (Proper) Riemann integral | 線上觀看 | |
| Class24 | 線上觀看 | |
| Class25 | 線上觀看 | |
| Class26 Sec. 2.13. Lebesgue decomposition | 線上觀看 | |
| Class27 Sec. 2.13. Lebesgue decomposition | 線上觀看 | |
| Class28 Sec. 2.14 Fundamental Thm of Calculus on | 線上觀看 | |
| Class29 | 線上觀看 | |
| Class30 | 線上觀看 | |
| Class31 | 線上觀看 | |
| Class32 | 線上觀看 | |
| Class33 | 線上觀看 |
課程目標
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
| 單元主題 | 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, 1982.
評分標準
| 項目 | 百分比 |
| 平時成績 | 1/3 |
| 期中考 | 1/3 |
| 期末考 | 1/3 |
本課程行事曆提供課程進度與考試資訊參考。
| 單元主題 | 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 |
