課程資訊
課程名稱
實分析二
Real Analysis (Ⅱ) 
開課學期
108-2 
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
 
學分
3.0 
全/半年
半年 
必/選修
必修 
上課時間
星期一3,4(10:20~12:10)星期三3,4(10:20~12:10) 
上課地點
新402新402 
備註
研究所基礎課。
總人數上限:60人
外系人數限制:15人 
Ceiba 課程網頁
http://ceiba.ntu.edu.tw/1082MATH7202_RA_II 
課程簡介影片
 
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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 

課程目標
This course aims to introduce basic theory and techniques of modern analysis. 
課程要求
Course prerequisite: Introduction to Mathematical Analysis I, II; Real Analysis I 
預期每週課後學習時數
 
Office Hours
 
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 
指定閱讀
Fon-Che Liu, Real Analysis, Oxford University Press  
評量方式
(僅供參考)
 
No.
項目
百分比
說明
1. 
mid-term exam 
30% 
 
2. 
final exam 
40% 
 
3. 
homework 
30% 
 
 
課程進度
週次
日期
單元主題
第3週
3/02,3/04  1. Introduction
2. Maximal functions and Vitali covering lemma 
第4週
3/09,3/11  1. Lebesgue differentiation Theorem
2. Lebesgue set of a function
3. Differentiation Theorem for a regularly-shrinking family of sets
4. Points of density
5. Rectifiable curves  
第5週
3/16,3/18  1. point of density
2. Functions of bounded variation (BV): total variation, positive variation, negative variation
3. A function of bounded variation is the difference of two increasing functions 
第6週
3/23,3/25  1. Vitali covering and Vitali Covering Lemma
2. An increasing function on [a,b] is differentiable almost every 
第7週
3/30,4/01  1. Differentiation of functions of bounded variation
2. Absolutely continuous and fundamental theorem of calculus
3. Curve length and absolutely continuous 
第8週
4/06,4/08  Basic principles of linear analysis:
1. Metric space, normed vector space, Banach space, bounded linear transformation
2. Baire category theorem 
第9週
4/13,4/15  1. Principle of Uniform Boundedness
2. Banach-Steinhaus Theorem
3. The inverse of I -T; log (I+T) 
第10週
4/20,4/22  1. Nowhere differentiable continuous functions
2. Open mapping theorem 
第11週
4/27,4/29  midterm examination 
第12週
5/04,5/06  Closed graph theorem; examples and counterexamples 
第13週
5/11,5/13  1. Axiom of choice, Zorn's lemma, Hausdorff maximality principle
2. Hahn-Banach Theorem 
第14週
5/18,5/20  1. Complex Hahn-Banach Theorem
2. Inner product space, Hilbert space
3. Parallelogram identity, Schwarz's inequality, triangle inequality
4. L2 space and completeness 
第15週
5/25,5/27  1. Space l ²
2. Orthogonal projection theorem
3. Riesz representation theorem
4. Lebesgue-Nikodym theorem 
第16週
6/01,6/03  1. Orthonormal family, Gram-Schmidt process, Hermite functions
2. Bessel's inequaity, Riesz-Fischer Theorem
3. Orthonormal basis and separability  
第17週
6/08,6/10  1. Fourier series, Dirichlet kernel, Fejer kernel
2. Lp space, Young's inequality, Holder inequality 
第18週
6/15,6/17  1. Completeness of Lp
2. The space lp
3. Signed measure: positive and negative sets, Hahn decomposition