課程名稱 |
實分析二 Real Analysis (Ⅱ) |
開課學期 |
104-2 |
授課對象 |
理學院 數學研究所 |
授課教師 |
陳俊全 |
課號 |
MATH7202 |
課程識別碼 |
221 U2880 |
班次 |
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學分 |
3 |
全/半年 |
半年 |
必/選修 |
選修 |
上課時間 |
星期一1,2(8:10~10:00)星期三3,4(10:20~12:10) |
上課地點 |
天數102天數102 |
備註 |
研究所基礎課。 總人數上限:60人 |
Ceiba 課程網頁 |
http://ceiba.ntu.edu.tw/1042MATH7202_RA2 |
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 |
課程大綱
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課程概述 |
In this semester, this course will cover the following contents.
1. More properties of L^p spaces.
2.Elements of Functional Analysis: Baire Category Theorem and its consequences, open mapping theorem and closed graph theorem, separation principles and Hahn-Banach theorm, Hilbert spaces
3. Abstract measure and integration theory: exterior measure, Caratheodory's theorem, extension theorem, integration on a measure space, product measure and Fubini's theorem, signed measure and absolutely continuity of measure.
4. Hausdorff measure |
課程目標 |
This course aims to introduce basic theory and techniques of modern analysis. |
課程要求 |
Real Analysis I |
預期每週課後學習時數 |
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Office Hours |
另約時間 |
指定閱讀 |
待補 |
參考書目 |
Textbooks:
[1] Elias M. Stein and Rami Shakarchi, Real Analysis
[2] Fon-Che Liu, Lecture notes in Real Analysis
Reference books:
[3] Richard Wheeden and Antoni Zygmund, Measure and Integral: An Introduction to Real Analysis
[4] Elliott H. Leib and Michael Loss, Analysis |
評量方式 (僅供參考) |
No. |
項目 |
百分比 |
說明 |
1. |
homework and quiz |
30% |
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2. |
mid-term exam |
30% |
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3. |
final exam |
40% |
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週次 |
日期 |
單元主題 |
第1週 |
2/22,2/24 |
Baire Category Theorem, nowhere differentiable functions |
第2週 |
2/29,3/02 |
Normed vector space, Banach space,
Principle of Uniform Boundedness, Banach-Steinhaus Theorem |
第3週 |
3/07,3/09 |
Completeness of L(X,Y) if Y is complete
Open mapping theorem,
Applications of open mapping theorem |
第4週 |
3/14,3/16 |
Closed graph theorem
lp space: Holder inequality, Minkowski inequality,
completeness |
第5週 |
3/21,3/23 |
Dual space
Dual space of Lp: Riesz representation theorem |
第6週 |
3/28,3/30 |
Inner product space and Hilber space
Parallelogram identity, Schwarz inequality, Triangle inequality
Complete orthonomal system, Bessel's inequality, Parseval's identity
A separable infinite dim Hilbert space is isometric to l2
Trigonometric functions form a complete orthonormal system on
[0,2Pi] |
第7週 |
4/04,4/06 |
Complete orthonormal system
Bessel's inequality
Parserval's identity
Isometry between a separable Hilbert space and L2 |
第8週 |
4/11,4/13 |
General measure space
Outer measure |
第9週 |
4/18,4/20 |
Caratheodory condition
Construction of a measure space |
第10週 |
4/25,4/27 |
Egorov's theorem
Fatou's lemma
Monotone convergence theorem
Dominated convergence theorem |
第11週 |
5/02,5/04 |
Monotone convergence theorem
Dominatd convergence theorem
General integration theory |
第12週 |
5/09,5/11 |
Unique extension from a premeasure to a measure |
第13週 |
5/16,5/18 |
Product measure |
第14週 |
5/23,5/25 |
Fubini's theroem
Polar coordinates |
第15週 |
5/30,6/01 |
Signed measure
Total variation of a signed measure
Hahn decomposition and Jordan decomposition |
第16週 |
6/06,6/08 |
Lp space
Lebesgue-Radon-Nikodym theorem |
第17週 |
6/13,6/15 |
Lebesgue-Radon-Nikodym theorem
Dual space of Lp space |
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