課程資訊
課程名稱
 分析導論一
Introduction to Mathematical Analysis(Ⅰ) 
開課學期
 109-1 
授課對象
 社會科學院  經濟學研究所  
授課教師
 陳俊全 
課號
 ECON5129 
課程識別碼
 323 U2030 
班次
  
學分
 5.0 
全/半年
 半年 
必/選修
 選修 
上課時間
 星期二2,3,4(9:10~12:10)星期四3,4(10:20~12:10) 
上課地點
  
備註
 上課資訊依課號MATH2213訊息為主。限選修ECON課號，方可認定為經濟系選修課。
限學士班三年級以上 或 限碩士班以上 或 限博士班
總人數上限：20人 
Ceiba 課程網頁
 http://ceiba.ntu.edu.tw/1091ECON5129_MA 
課程簡介影片
 
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課程概述

這門課是數學系的入門課程，主要是讓學生熟悉數學分析的語言，訓練學生更嚴謹的數學證明邏輯，也是更高階分析課程的基礎。為了要有更廣的觀點，我們會從基本的點集拓樸切入，引進極限的觀念，隨後介紹連續及微分，還有這些觀念的相關定理及應用，而後將介紹積分及相關的課題，如果時間允許，我們也會涉略基本的測度論。 

課程目標
 讓學生熟悉數學分析的語言，能夠使用分析的工具操作嚴謹的證明。 
課程要求
 週作業，期中考，期末考。
預備知識: calculus, linear algebra 
預期每週課後學習時數
  
Office Hours
 每週四 13:00~15:00 備註： 週四：王舜傑(天數455) 
指定閱讀
 待補 
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
3. Mathematical Analysis. Second Edition. Tom M. Apostol.
4. William R. Wade, An Introduction to Analysis, Prentice Hall, 4th 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1週
 9/15,9/17  0. Introduction -problems arising from calculus; new topics:
0.1.real numbers and completeness
0.2 what is infinity?
0.3 topology of the Euclidean space: Riemann integral and compactness
0.4 uniform convergence of functions
0.5 differentiation in R^n
0.6 Solve a system of non-linear equations:- Inverse and Implicit Function Theorems
0.7 Lebesgue's Theorem for integrals
0.8 Fourier series

1. The real number system and the Euclidean space
1.1 Sets and Functions:
- power set of A, product of A and B
- domain, target, range of a function, one-to-one, onto
1.2 Origin of number concept
- Piraha people in the Amazon rainforest
- Research on infants
1.3 Number system: natural numbers, integers, rational numbers 
第2週
 9/22,9/24  1.4 Ordered Fields
- addition axioms, multiplication axioms and order axioms
- sequence and limit: uniqueness of limits, sandwich lemma, limits of a
sum and a product
- Cauchy sequence
- Axiom of completeness  
第3週
 9/29,10/01  - Basic properties of Cauchy sequences
- Axioms of a complete ordered field  
第4週
 10/06,10/08  1-5 Construction of a complete ordered field
1-5-1 three approaches: infinite decimals, Cauchy sequences and Dedekind cuts
1-5-2 Cauchy sequence approach:
- S=the set of all rational Cauchy sequences
- an equivalence relation on S and the corresponding equivalence classes
- addition and multiplication on the equivalence classes 
第5週
 10/13,10/15  1-5-2 Cauchy sequence approach:
- order on the equivalence classes
- Cauchy sequences in the space of the equivalence classes
-the equivalent classes together with the addition, multiplication and order forms a complete ordered field 
第6週
 10/20,10/22  -Theorem: There exists a "unique" complete ordered field, called the real number system.
- Monotone sequence property (MSP)
-sup, inf and the least upper bound property (LUBP)
 
第7週
 10/27,10/29  -Theorem: The three versions of completeness (CSP)+(AP), (MSP) and (LUBP) are equivalent.
1-6 limsup and liminf 
第8週
 11/03,11/05  - more properties and applications of limsup and liminf
1-7 Cantor's theory of infinity
- Definition of card A=card B and card A<card B
- finite, countable and uncountable
- an infinite subset of a countable set is countable
- card N = card Q < card R = card RxR=card P(N), Cantor's diagonal method
- card A < card P(A)
- existence of an algebraic number 
第9週
 11/10,11/12  - Schroder-Bernstein Theorem
- continuum hypothesis: Godel and Cohen
1-8 Some "paradoxes" about real numbers
- a number of all knowledge
- Pi is a normal number?
Borel's theorem: Almost every real number is normal.
- Richard's paradox
1-9 Complex numbers
1-10 Euclidean space
- norm, metric, inner product, Schwarz's inequality

Chapter 2 Topologies of Metric Spaces
2-1 Metric space: definition and examples
 
第10週
 11/17,11/19  Midterm examination
2-2 Open sets and interior of a set
 
第11週
 11/24,11/26  2-3 Closed sets, accumulation points, closure of a set
2-4 Boundary of a set
2-5 Sequences and limits
2-6 Completeness of a metric space
 
第12週
 12/01,12/03  Chapter 3 Compact sets
3-1. Examples: the difference between I= [0,1] and I=(0,1]; consider continuous function on I
3-2 Sequentially compact: bisection process and bounded sequence; Heine-Borel Theorem
3-3 Open cover and compact:
- examples
 
第13週
 12/08,12/10  - compact implies bounded and closed; counterexample
- totally bounded;
- Bolzano-Weierstrass Theorem
 
第14週
 12/15,12/17  - compact iff totally bounded and complete in a metric space
3-4 Path-connected and connected
- path connected implies connected

Chapter 4 Continuous maps
4-1 Continuity
- limit at a point
- continuous at a point and on the whole domain
- continuity defined by sequential limits 
第15週
 12/22,12/24  - continuity characterized by preimages of open and closed sets
- continuity for +,-,×,÷, and f(g(x))
4-2 Images of compact and connected sets
4-3 Real-valued functions
- Maximum-minimum theorem
- Intermediate value theorem
4-4 Uniform continuity 
第16週
 12/29,12/31  Chapter 5 Uniform convergence of functions
-Motivations
5-1 Pointwise and uniform convergence
- examples
- uniform convergence implies pointwise convergence
- uniform convergence iff sup ρ(f_k,f) → 0
- Theorem: The limit function of an uniformly convergent sequence
of continuous functions is continuous.
5-2 Cauchy criterion and M test
- Cauchy criterion and uniform convergence
- examples: uniformly convergence of series of functions 
第17週
 1/05,1/07  5-3 Integration and differentiation of sequences and series of
functions
- Theorem: uniform convergence implies convergence of the
integrals
- Theorem : pointwise convergence of the functions and uniform
convergence of their derivatives together imply differentiability of
the limit function
5-4 The space of continuous functions
- completeness property
- equicontinuity
- Arzela-Ascoli Theorem