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課程名稱 |
實分析二
Real Analysis (Ⅱ) |
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開課學期 |
111-2 |
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授課對象 |
理學院 應用數學科學研究所 |
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授課教師 |
陳俊全 |
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課號 |
MATH7202 |
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課程識別碼 |
221 U2880 |
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班次 |
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學分 |
3.0 |
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全/半年 |
半年 |
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必/選修 |
選修 |
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上課時間 |
星期一3,4(10:20~12:10)星期三3,4(10:20~12:10) |
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上課地點 |
新202天數202 |
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備註 |
總人數上限：40人 |
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課程簡介影片 |
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核心能力關聯 |
核心能力與課程規劃關聯圖 |
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課程大綱
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為確保您我的權利,請尊重智慧財產權及不得非法影印
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課程概述 |
1. Differentiation : Hardy-Littlewood maximal function, Lebesgue differentiation theorem, functions of bounded variation, absolutely continuous functions, differentiability of functions
2.Elements of Functional Analysis: Baire Category Theorem and its consequences, open mapping theorem and closed graph theorem, separation principles and Hahn-Banach theorem, Hilbert spaces, Banach spaces, dual spaces
3. L^p spaces
4. Signed Measures: absolute continuity, Radon-Nikodym Theorem
5. Fourier Transform
6. Hausdorff Measure and Fractals |
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課程目標 |
This course aims at introducing basic theory and techniques of modern analysis. |
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課程要求 |
Course prerequisite: Introduction to Mathematical Analysis I, II; Real Analysis I |
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預期每週課後學習時數 |
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Office Hours |
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指定閱讀 |
Elias M. Stein and Rami Shakarchi, Real Analysis
Fon-Che Liu, Real Analysis, Oxford University Press |
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參考書目 |
[1] Elias M. Stein and Rami Shakarchi, Real Analysis
[2] Richard Wheeden and Antoni Zygmund, Measure and Integral: An Introduction to Real Analysis
[3] Elliott H. Leib and Michael Loss, Analysis |
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評量方式
(僅供參考) |
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No. |
項目 |
百分比 |
說明 |
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1. |
mid-term exam |
30% |
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2. |
final exam |
40% |
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3. |
homework |
30% |
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週次 |
日期 |
單元主題 |
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第1週 |
2/20-23 |
0. Introduction
1. Differentiation:
1-1 Differentiation of the integral: maximal functions,
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第2週 |
2/27-30 |
Vitali covering lemma for finitely many balls |
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第3週 |
3/06-09 |
properties of maximal functions, Lebesgue Differentiation Theorem, point of Lebesgue density, Lebesgue set of a function, shrink regularly |
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第4週 |
3/13-16 |
1-3 Differentiability of functions: rectifiable curves, functions of bounded variation (BV functions), total variation, positive variation, negative variation, Vitali covering lemma for infinitely many balls |
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第5週 |
3/20-23 |
Covering lemma for Vitali-sense coverings, the derivative of an increasing function exists a.e. |
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第8週 |
4/10-12 |
2. Basic Principles of Functional Analysis
2-1. Baire Category Theorem and Banach-Steinhaus Theorem |
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第9週 |
4/17-19 |
2-2. Open Mapping and Closed Graph Theorems |
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第10週 |
4/24-26 |
2-3. Hausdorff Maximality Principle and Axiom of Choice |
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第11週 |
5/01-03 |
2-4. Hahn-Banach Theorem and its Applications |
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第12週 |
5/08-10 |
2-5. Hilbert Space and Projection Theorem |
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第13週 |
5/15-17 |
3. Abstract Measure and Integration Theory
3-1. Measure Space
3-2. Premeasure and Extension |
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第14週 |
5/22-24 |
3-3. Integration
3-4. Product measures |
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第15週 |
5/29-31 |
3-5. Tonelli and Fubini Theorems |
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第16週 |
6/05-07 |
3-6 Signed measure: Hahn decomposition, Jordan decomposition
Final Examination |
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第17週 |
6/12-14 |
3-7 Radon-Nikodym Theorem
4. Lp space |
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