Type of Credit: Partially Required
Credit(s)
Number of Students
Basic functional analysis: Normed linear space, Banach space, Hilbert space, bounded linear operator, Baire category theorem, Hahn-Banach extension theorem, open mapping theorem, closed graph theorem, uniformly bounded principle.
Lp-space: Definition of Lp space, Young's inequality, Holder's inequality, Minkowski's inequality, Completeness of Lp space
Modes of convergence: Convergence almost everywhere, almost uniform convergence, convergence in measure, convergence in Lp, Relation between various types of convergence
Signed measure: Definition of signed measure, Hahn and Jordan decomposition, positive/negative/total variation, integration with respect ot signed measure
Radon-Nikodym theorem and Riesz representation theorem: Radon-Nikodym theorem, Riesz representation theorem for Lp-space
Product measure: Product σ-algebra, Product measure, Tonelli and Fubini's theorem, Surface measure on Sn-1
Differnetiation: Differnetiation of monotone function, Lebesgue differnetiation theorem, functions of bounded variation, differnetiation of an indefinite integral
能力項目說明
本課程之目標在建立學生分析的基礎作為未來相關領域之發展
| 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 |
Integration |
Integration of General Measurable Functions and their Properties, Lebesgue Dominated Convergence Theorem |
演練與講解;學習評量測驗 |
3.0 |
4.5 |
|
2 |
Integration |
Lebesgue-Stieltjes Integral, Reduction Integral over R, Point-Mass Distribution |
演練與講解;學習評量測驗 |
3.0 |
4.5 |
|
3 |
Basic Functional Analysis |
Normed Linear Space, Banach Space, Hilbert Space, Bounded Linear Operator, Baire Category Theorem |
演練與講解;學習評量測驗 |
3.0 |
4.5 |
|
4 |
Basic functional analysis |
Hahn-Banach extension theorem, open mapping theorem, closed graph theorem, uniformly bounded principle |
演練與講解;學習評量測驗 |
3.0 |
4.5 |
|
5 |
Lp-space |
Definition of Lp space, Young's inequality, Holder's inequality |
演練與講解;學習評量測驗 |
3.0 |
4.5 |
|
6 |
Lp-space |
Minkowski's inequality, Completeness of Lp space |
演練與講解;學習評量測驗 |
3.0 |
4.5 |
|
7 |
Modes of convergence |
Convergence almost everywhere, almost uniform convergence, convergence in measure |
演練與講解;學習評量測驗 |
3.0 |
4.5 |
|
8 |
Modes of convergence |
convergence in Lp, Relation between various types of convergence |
演練與講解;學習評量測驗 |
3.0 |
4.5 |
|
9 |
Mideterm |
Midterm |
Midterm |
0.0 |
0.0 |
|
10 |
Signed measure |
Definition of signed measure, Hahn and Jordan decomposition |
演練與講解;學習評量測驗 |
3.0 |
4.5 |
|
11 |
Signed measure |
positive/negative/total variation, integration with respect ot signed measure |
演練與講解;學習評量測驗 |
3.0 |
4.5 |
|
12 |
Radon-Nikodym theorem and Riesz representation theorem |
Radon-Nikodym theorem |
演練與講解;學習評量測驗 |
3.0 |
4.5 |
|
13 |
Radon-Nikodym theorem and Riesz representation theorem |
Riesz representation theorem for Lp-space |
演練與講解;學習評量測驗 |
3.0 |
4.5 |
|
14 |
Product measure |
Product σ-algebra, Product measure |
演練與講解;學習評量測驗 |
3.0 |
4.5 |
|
15 |
Product measure |
Tonelli and Fubini's theorem, Surface measure on Sn-1 |
演練與講解;學習評量測驗 |
3.0 |
4.5 |
|
16 |
Differnetiation |
Differnetiation of monotone function, Lebesgue differnetiation theorem |
演練與講解;學習評量測驗 |
3.0 |
4.5 |
|
17 |
Differnetiation |
functions of bounded variation, differnetiation of an indefinite integral |
演練與講解;學習評量測驗 |
3.0 |
4.5 |
|
18 |
Final |
Final |
Final |
0.0 |
0.0 |
成績評量:
期中考成績: 40%
期末考成績：60%
學生有任何建議可於開學上課時提出。
Note on measure theory by M. Papadimitrakis
References:
1. Real Analysis, 3rd edition, by H.L. Royden.
2. Principles of Real Analysis, 2nd edition, by C. D. Aliprantis and O. Burkinshaw.
3. Measure and Integral by R. L. Wheeden and A. Zygmund.
4. The Elements of Integration and Lebesgue Measure by R. G. Bartle
| 書名 Book Title | 作者 Author | 出版年 Publish Year | 出版者 Publisher | ISBN | 館藏來源* | 備註 Note |
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