Type of Credit: Elective
Credit(s)
Number of Students
-Topological Vector Spaces:
Basic poperties of topological vector spaces, Separation properties, Linear mappings, Finite-dimensional spaces, Metrization, Boundedness and continuity, Seminorms and local convexity, Quotient spaces, Examples
-Completeness:
Baire category, The Banach-Steinhaus Theorem, The open mapping theorem, The closed graph theorem, Bilinear mappins
-Convexity:
The Hahn-Banach theorems, Weak topologies, Compact convex sets, Vector-valued integration, Holomorphic functions
-Duality in Banach Spaces:
The normed dual of a normed space, Adjoints, Compact operators
-Some Applcations
A continuity theorem, Closed subspaces of L^p-spaces, The range of vector-valued measure, A generalized Stone-Weierstrass theorem, Two interpolation theorems, A fixed point theorem, Haar measure on compact groups, Uncomplemented subspaces
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 |
Topological Vector Spaces |
Basic poperties of topological vector spaces, Separation properties, Linear mappings |
習題討論 |
3.0 |
6.0 |
|
2 |
Topological Vector Spaces |
Finite-dimensional spaces, Metrization, Boundedness and continuity |
習題討論 |
3.0 |
6.0 |
|
3 |
Topological Vector Spaces |
Seminorms and local convexity, Quotient spaces, Examples |
習題討論 |
3.0 |
6.0 |
|
4 |
Completeness |
Baire category, The Banach-Steinhaus Theorem |
習題討論 |
3.0 |
6.0 |
|
5 |
Completeness |
The open mapping theorem |
習題討論 |
3.0 |
6.0 |
|
6 |
Completeness |
The closed graph theorem, Bilinear mappins |
習題討論 |
3.0 |
6.0 |
|
7 |
Convexity |
The Hahn-Banach theorems |
習題討論 |
3.0 |
6.0 |
|
8 |
Convexity |
Weak topologies, Compact convex sets |
習題討論 |
3.0 |
6.0 |
|
9 |
Mideterm |
Midterm |
Midterm |
0.0 |
0.0 |
|
10 |
Convexity |
Vector-valued integration, Holomorphic functions |
習題討論 |
3.0 |
6.0 |
|
11 |
Duality in Banach Spaces |
The normed dual of a normed space, Adjoints |
習題討論 |
3.0 |
6.0 |
|
12 |
Duality in Banach Spaces |
Compact operators |
習題討論 |
3.0 |
6.0 |
|
13 |
Some Applcations |
A continuity theorem, Closed subspaces of L^p-spaces |
習題討論 |
3.0 |
6.0 |
|
14 |
Some Applcations |
The range of vector-valued measure, A generalized Stone-Weierstrass theorem |
習題討論 |
3.0 |
6.0 |
|
15 |
Some Applcations |
A fixed point theorem |
習題討論 |
3.0 |
6.0 |
|
16 |
Some Applcations |
Haar measure on compact groups |
習題討論 |
3.0 |
6.0 |
|
17 |
Some Applcations |
Uncomplemented subspaces |
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
無