Type of Credit: Elective
Credit(s)
Number of Students
-Test Functions and Distributions:
Test Function Spaces, Calculus with Distributions, Support of Distributions, Convolutions.
-Fourier Transform:
Basic Properties, Temper Distributions, Paley-Wiener Theorems, Sobolev's Lemma.
-Applications to Differential Equations:
Fundamental Solutions, Elliptic Equations.
-Commutative Banach Algebras:
Banach Algebras, Complex Homomorphism, Basic Properties of Spectra, Ideals and Homomorphisms, Gelfand Transforms.
-Bounded Operators on Hilbert Spaces:
Bounded Operators, A Commutativity Theorem, Resolutions of the Identity, The Spectral Theorem, Eigenvalues of Normal Operators.
能力項目說明
本課程之目標在讓學生了解如何利用學過的知識來探討無窮維泛函空間的性質，將其應用到偏微分方程、最佳化理論等相關領域。
| 教學週次Course Week | 彈性補充教學週次Flexible Supplemental Instruction Week | 彈性補充教學類別Flexible Supplemental Instruction Type |
|---|---|---|
| 週次Week | 課程主題Course Theme | 課程內容與指定閱讀Content and Reading Assignment | 教學活動與作業Activity and Homework | 學習投入時數Estimated time devoted to coursework per week | |
|---|---|---|---|---|---|
| 課堂講授Lecture Hours | 課程前後Preparation Time | ||||
|
1 |
Test Functions and Distributions | Test Function Spaces, Calculus with Distributions |
習題討論 |
3.0 |
6.0 |
|
2 |
Test Functions and Distributions | Support of Distributions |
習題討論 |
3.0 |
6.0 |
|
3 |
Test Functions and Distributions | Convolutions |
習題討論 |
3.0 |
6.0 |
|
4 |
Fourier Transform | Basic Properties |
習題討論 |
3.0 |
6.0 |
|
5 |
Fourier Transform | Temper Distributions |
習題討論 |
3.0 |
6.0 |
|
6 |
Fourier Transform | Paley-Wiener Theorems |
習題討論 |
3.0 |
6.0 |
|
7 |
Fourier Transform | Sobolev's Lemma |
習題討論 |
3.0 |
6.0 |
|
8 |
Applications to Differential Equations | Fundamental Solutions |
習題討論 |
3.0 |
6.0 |
|
9 |
Mideterm |
Midterm |
Midterm |
0.0 |
0.0 |
|
10 |
Applications to Differential Equations | Elliptic Equations |
習題討論 |
3.0 |
6.0 |
|
11 |
Commutative Banach Algebras | Banach Algebras, Complex Homomorphism |
習題討論 |
3.0 |
6.0 |
|
12 |
Commutative Banach Algebras | Basic Properties of Spectra, Ideals and Homomorphisms |
習題討論 |
3.0 |
6.0 |
|
13 |
Commutative Banach Algebras | Gelfand Transforms |
習題討論 |
3.0 |
6.0 |
|
14 |
Bounded Operators on Hilbert Spaces | Bounded Operators, A Commutativity Theorem |
習題討論 |
3.0 |
6.0 |
|
15 |
Bounded Operators on Hilbert Spaces | Resolutions of the Identity |
習題討論 |
3.0 |
6.0 |
|
16 |
Bounded Operators on Hilbert Spaces | The Spectral Theorem |
習題討論 |
3.0 |
6.0 |
|
17 |
Bounded Operators on Hilbert Spaces | Eigenvalues of Normal Operators |
習題討論 |
3.0 |
6.0 |
|
18 |
Final |
Final |
Final |
0.0 |
0.0 |
習題演練 50%
期末口試 50%
Text: Functional Analysis, Walter Rudin
Reference: Functional Analysis, Yosida
| 書名 Book Title | 作者 Author | 出版年 Publish Year | 出版者 Publisher | ISBN | 館藏來源* | 備註 Note |
|---|