課程資訊
課程名稱
 實分析一
Real Analysis (Ⅰ) 
開課學期
 104-1 
授課對象
 理學院  應用數學科學研究所  
授課教師
 陳俊全 
課號
 MATH7201 
課程識別碼
 221 U2870 
班次
  
學分
 3 
全/半年
 半年 
必/選修
 必修 
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
 總人數上限：50人 
Ceiba 課程網頁
 http://ceiba.ntu.edu.tw/1041MATH7201_RA 
課程簡介影片
 
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課程概述

The course will cover Chapters 1-4 of [1] and some contents of [2], including
1.Measure Theory: outer measure, Caratheodory outer measures, n-dimensional Lebesgue measure
2.Integration Theory: measurable functions, Lebesgue integral, monotone convergence and Lebesgue dominated convergence theorem, Fubini’s theorem
3.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
4.Lp-spaces
5.Differentiation and Integration: Hardy-Littlewood maximal function, Lebesgue differentiation theorem, functions of bounded variation, absolutely continuous functions, differentiability of functions
 

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 
預期每週課後學習時數
  
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  
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
 9/14,9/16  Motivations for the Lebesgue theory: Limits of Riemann integrals; Fourier series; Measure=length, area, volume
Axioms of measure
Background (1870-1910): Weierstrass's nowhere differential function, Peano's space-filling curve 
第2週
 9/21,9/23  Lemma for rectangles and cubes;
Outer measure (exterior measure);
Outer Jordan content;
"closed cubes" can be replaced by "open cubes" or "rectangles" in the definition of outer measure;
Outer measure of a cube/rectangle;
Properties of outer measure: outer measure of empty set, monotonicity, countable sub-additivity, sets separated by a positive distance, set can be written as countable union of almost disjoint cubes 
第3週
 9/28,9/30  Properties of open sets;
Lebesgue measurable sets. 
第4週
 10/05,10/07  Properties of measurable sets
Countable Additivity  
第5週
 10/12,10/14  sigma algebra;
Borel sets;
Non-measurable set;
Invariance properties of measurable sets: translation, rotation, reflection and linear transformation 
第6週
 10/19,10/21  Measurable functions
Approximation by simple functions and step functions 
第7週
 10/26,10/28  Egorove's theorem
Lusin's theorem
Convergence in measure 
第8週
 11/02,11/04  Integration theory
Lebesgue integral: simple functions, bounded functions, nonnegative functions, general case
Bounded convergence theorem
Fatou's lemma  
第9週
 11/09,11/11  Bounded convergence theorem, Monotone convergence theorem, Fatou's lemma, Dominated convergence theorem 
第10週
 11/16,11/18  L1 space, completeness of L1 space;
Linear transformation and integral;
L1 continuity under translation 
第11週
 11/23,11/25  Fubini's theorem and Tonelli's theorem 
第12週
 11/30,12/02  Hardy-Littlewood maximal function, simple Vitali lemma;
Lebesgue differentiation theorem 
第13週
 12/07,12/09  Points of Lebesgue density, Lebesgue points of a function, differentiation on sets shrinking regularly;
Function of bounded variation 
第14週
 12/14,12/16  Good kernels and approximations to the identity 
第15週
 12/21,12/23  Simple Vitali lemma II; Vitali covering lemma 
第16週
 12/28,12/30  Differentiability of monotone functions and BV functions,
Absolutely continuous functions 
第17週
 1/04,1/06  Lp space: Lp and L-infinite norm
Young's inequality
Holder and Minkowski inequalities
Completeness of Lp space