課程資訊
課程名稱
 實分析一
Real Analysis (Ⅰ) 
開課學期
 108-1 
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
 3.0 
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
 天數102天數102 
備註
 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/1081MATH7201_RA_I 
課程簡介影片
 
核心能力關聯
 核心能力與課程規劃關聯圖
課程大綱
為確保您我的權利,請尊重智慧財產權及不得非法影印
課程概述

Real Analysis is indispensable for in-depth understanding and effective application of methods of modern analysis. The course will cover the following topics:
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 theorem, Hilbert spaces
4. 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
 Fon-Che Liu, Real Analysis, Oxford University Press 
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 
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1週
 9/09,9/11  Introduction:
1. Bad functions: Weierstrass' nowhere differentiable function; Peano's space-filling function
2. Limits of integrals and Fourier series
Volume of rectangles; open sets and cubes; Jordan outer measure and Lebesgue outer measure; 
第2週
 9/16,9/18  Properties of Lebesgue outer measure: monotonicity, countable sub-additivity, etc. 
第3週
 9/23,9/25  Measurable sets: open sets; closed sets; unions, intersections, and complements of measurable sets  
第4週
 9/30,10/02  Measurable sets: additivity for disjoint sets; limit property; translation and dilation properties of Lebesgue measure 
第5週
 10/07,10/09  1. σ algebra, Borel sets, axiom of choice, non-measurable sets
2. Definition of measurable functions 
第6週
 10/14,10/16  1. Properties of measurable functions: sum and product, composition with a continuous function, sup/inf and limsup/liminf, lim
2. Approximations by simple and step functions, Egorov's Theorem 
第7週
 10/21,10/23  1. Lusin's Theorem; Littlewood's three principles
2. Integration theory: integration of simple functions; integration of bounded functions supported on finite-measure sets 
第8週
 10/28,10/30  1. Bounded convergence theorem
2. Integration of non-negative functions: linearity, additivity, monotonicity, and other basic properties 
第9週
 11/04,11/06  1. General case: integrable functions, Dominated Convergence Theorem
2. Midterm examination 
第10週
 11/11,11/13  1. Invariance properties of integrals: translation, reflection, dilation; absolute continuity of an integrable function
2. L^1 space, completeness of L^1 space 
第11週
 11/18,11/20  Fubini's Theorem 
第12週
 11/25,11/27  1. Tonelli's Theroem; Applications of Fubini's Theorem
2. Abstract measure space 
第13週
 12/02,12/04  1. Construction of outer measures
2. Caratheodory measurable sets
3. Equivalence of Caratheodory measurability and Lebesgue measurability in Euclidean spaces 
第14週
 12/09,12/11  1. Premeasure and extension theorem
2. Integration theory on an abstract measure space: measurable function, Egorov's theorem, Fatou's Lemma, Monotone Convergence Theorem, Dominated Convergence Theorem 
第15週
 12/16,12/18  1. Product measure
2. Completion of the product measure of two Lebesgue measures 
第16週
 12/23,12/25  General Tonelli's theorems and Fubini's theorems 
第17週
 12/30,1/01  Polar coordinates and the corresponding product measure