課程資訊
課程名稱
 應用分析一
Applied Analysis (Ⅰ) 
開課學期
 110-1 
授課對象
 理學院  數學研究所  
授課教師
 陳俊全 
課號
 MATH7415 
課程識別碼
 221 U5740 
班次
  
學分
 3.0 
全/半年
 半年 
必/選修
 選修 
上課時間
 星期二3,4,5(10:20~13:10) 
上課地點
 天數202 
備註
 總人數上限：40人 
Ceiba 課程網頁
 http://ceiba.ntu.edu.tw/1101MATH7415_aa1 
課程簡介影片
 
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課程概述

This course focuses on the most fundamental theory and concepts in mathematical analysis.  

課程目標
 The course will cover the following topics:
1. continuity, compactness, uniform convergence and their applications.
2.Measure Theory: outer measure, Caratheodory outer measures, n-dimensional Lebesgue measure
3.Integration Theory: measurable functions, Lebesgue integral, monotone convergence and Lebesgue dominated convergence theorem
4.Elements of Functional Analysis: Baire Category Theorem and its consequences, open mapping theorem and closed graph theorem, Hahn-Banach theorem, Hilbert spaces 
課程要求
 待補 
預期每週課後學習時數
  
Office Hours
  
指定閱讀
 1. Elias M. Stein and Rami Shakarchi, Real Analysis
2. Richard Wheeden and Antoni Zygmund, Measure and Integral: An Introduction to Real Analysis 
參考書目
 1. Jerrold E. Marsden and Michael J. Hoffman, Elementary Classical Analysis, 2nd Edition
2. Walter Rudin, Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics), McGraw-Hill Education; 3rd edition 
評量方式
(僅供參考)
  
No.
項目
百分比
說明
1. 
 homework and quiz 
25% 
  
2. 
 midterm exam 
35% 
  
3. 
 final exam 
40% 
  
 
課程進度
週次
日期
單元主題
第2週
 9/28  Part 1 Introduction
Goal, Background, and Motivation
Related Problems:
1. Length, area, volume
2. Fourier series
3. Limits of integrals

 
第3週
 10/05  Part 2 Fundamental Concepts of Analysis
2-1 The Real Number System
2-1-2 Ordered fields: axiomatic method
field axioms, order axioms
2-1-2 Completeness of the real numbers
sequence, limit, monotone sequence property,
definition of the real number system 
第4週
 10/12  applications of monotone sequence property, Archimedean property,
non-Archimedean ordered field, infinitesimal  
第5週
 10/19  2-1-3 Cauchy Sequence
properties of Cauchy sequences, CSP(Cauchy sequence
property), bisection process
Theorem MSP iff CSP+Archimedean Property 
第6週
 10/26  2-1-4 Sup and Inf
maximum and minimum, supremum and infimum
Theorem (MSP) iff (CSP)+ (Arch P) iff (LUBP)
cluster point, properties of limsup and liminf  
第7週
 11/02  2-2 Cantor’s Theory of Infinity
cardinality and bijection, countable,
the set of real numbers is uncountable (Cantor's diagonal
argument),
card (A)< card (power set of A)
2-3 Topology of the Eulidean spaces
open sets and interior points 
第8週
 11/09  2-3 Topology of the Eulidean spaces
limits in the Euclidean spaces
open sets and interior points
closed sets and closure
boundary set
2-4 Compact sets
continuous function
 
第9週
 11/16  2-4 Compact sets
2-4-1 continuous function
2-4-2 sequentially compact:
Heine Borel Theorem,
existence of a maximum for continuous functions
2-4-3 open covers and compactness
Bolzano-Weierstrass Theorem 
第10週
 11/23  Midterm examination 
第11週
 11/30  2-4-3 open covers and compactness
Bolzano-Weierstrass Theorem
2-4-4
Comparison between sequentially compact and compact
Part 3 Measure theory
3-1 Introduction: motivation, integral and area, properties of area
3-2 Jordan measure (Jordan content)
Definition
Q: What is the length of rational numbers between 0 and 1?
3-3 Lebesgue exterior measure
Definition
Examples 
第12週
 12/07  3-3 Lebesgue exterior measure
Monotonicity, countable sub-additivity, additivity of sets with positive distance, approximation by open sets  
第13週
 12/14  3-4 Measurable sets
Motivation of the definition, measure zero sets, measurability of closed sets and compliment sets 
第14週
 12/21  3-4 Measurable sets
countable additivity of measurables
3-5 sigma-algebra
Borel sigma-algebra, Borel set, G-delta set, F-sigma set, sigma-algebra of Lebesgue measurable set= completion of Borel sigma-algebra, 
第15週
 12/28  3-6 Invariance properties of Lebesgue measure
3-7 Construction of a non-measurable set
Axiom of Choice
Part 4 Measurable functions
4-1 Introduction  
第16週
 1/04  4-2 Definition and basic properties:
equivalent definitions, continuous function, composition of functions,
limit, sum and product, almost everywhere 
第17週
 1/11  4-3 Approximation by simple and step functions
4-4 Littlewood’s three principles: Egorov's Theorem, Lusin's Theorem
5. Integration Theory: a brief introduction
Integrals of simple functions, bounded functions, non-negative functions, and general functions.
Bounded convergence theorem, monotone convergence theorem, dominated convergence theorem, Fatou's lemma 
第18週
 1/18  Final examination