實變函數論(二)
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：
- H. L. Royden, Real analysis, third edition,Macmillan, 1989.
- A. Torchinsky, Real variables, Addison-Wesley, 1988.
- J. B. Conway, A course in functional analysis, 2nd edition, Springer, 1990
為求學習成效完美，請購買課本！
| 授課老師 | 應用數學系 吳培元老師 |
|---|---|
| 課程學分 | 3學分 |
| 授課年度 | 99學年度 |
| 授課對象 | 碩士班學生 |
| 預備知識 | Advanced calculus and general mathematical maturity. |
| 課程提供 | 課程影音 課程綱要 課程行事曆 |
| 週次 | 課程內容 | 課程影音 |
|---|---|---|
| 第三章 Metric spaces Class36 | 線上觀看 | |
| Class37 | 線上觀看 | |
| Class38 Sec3.4 Complete metric spaces | 線上觀看 | |
| Class39 Sec3.5 Compact metric spaces | 線上觀看 | |
| Class40 | 線上觀看 | |
| Class41 Sec3.7 Stone-Weierstiass Thm. | 線上觀看 | |
| Class42-43 | 線上觀看 | |
| 第四章 Banach spaces Class44 | 線上觀看 | |
| Class45 Sec. 4.2 Subspace & bases | 線上觀看 | |
| Class46 Sec. 4.4 Linear Transformations | 線上觀看 | |
| class47 Sec. 4.5 Principle of uniform bddness (Banach- Steinhaus Thm) | 線上觀看 | |
| Class48 | 線上觀看 | |
| Class49 | 線上觀看 | |
| Class50 | 線上觀看 | |
| Class51 | 線上觀看 | |
| Class52 | 線上觀看 | |
| Class53 | 線上觀看 | |
| Class54-55 | 線上觀看 | |
| Class56 | 線上觀看 | |
| Class57 | 線上觀看 | |
| Class58 | 線上觀看 | |
| Class59 | 線上觀看 | |
| Class60-61 Sec. 4.13 Adjoint operators | 線上觀看 | |
| Class62 | 線上觀看 | |
| Class63 | 線上觀看 | |
| Class64-65 | 線上觀看 | |
| Class66 sec. 5.2 Fredholm-Riesz-Schauder Theory | 線上觀看 | |
| Class67 | 線上觀看 | |
| Class68 | 線上觀看 | |
| Class69 Sec.5.3 Spectral theory | 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 |
| Functional analysis in Banach spaces | Principle of uniform boundedness, open mapping theorem, closed graph theorem, Hahn-Banach theorem, completeness of Lp-spaces |
| Compact operators | Fredholm-Riesz-Schauder theory for compact operators |
| Spectral theory of self-adjoint operators | Self-adjoint operators, normal operators |
| Hilbert spaces | Basic properties of Hilbert spaces |
課程書目
H. L. Royden, Real analysis, third edition,Macmillan, 1989.
A. Torchinsky, Real variables, Addison-Wesley, 1988.
J. B. Conway, A course in functional analysis, 2nd edition, Springer, 1990.
評分標準
| 項目 | 百分比 |
| 作業 | 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 |
| Functional analysis in Banach spaces | Principle of uniform boundedness, open mapping theorem, closed graph theorem, Hahn-Banach theorem, completeness of Lp-spaces |
| Compact operators | Fredholm-Riesz-Schauder theory for compact operators |
| Spectral theory of self-adjoint operators | Self-adjoint operators, normal operators |
| Hilbert spaces | Basic properties of Hilbert spaces |
