課程資訊
課程名稱
實分析二
Real Analysis (Ⅱ) 
開課學期
104-2 
授課對象
理學院  數學系  
授課教師
陳俊全 
課號
MATH7202 
課程識別碼
221 U2880 
班次
 
學分
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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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
 
Office Hours
另約時間 
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  
指定閱讀
待補 
評量方式
(僅供參考)
 
No.
項目
百分比
說明
1. 
homework and quiz 
30% 
 
2. 
mid-term exam 
30% 
 
3. 
final exam 
40% 
 
 
課程進度
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