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※ 本文是否可提供台大同学转作其他非营利用途？（须保留原作者 ID）
（是／否／其他条件）：
哪一学年度修课：
ψ 授课教师 (若为多人合授请写开课教师，以方便收录)
陈俊全
λ 开课系所与授课对象 (是否为必修或通识课 / 内容是否与某些背景相关)
数学系
δ 课程大概内容
第1周 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 The
orems
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周 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周 9/29,10/01
Basic properties of Cauchy sequences
Axioms of a complete ordered field
第4周 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周 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 f
orms a complete ordered field.
第6周 10/20,10/22
-Theorem: There exists a "unique" complete ordered field, called the real numb
er system.
- Monotone sequence property (MSP)
-sup, inf and the least upper bound property (LUBP)
第7周 10/27,10/29
-Theorem: The three versions of completeness (CSP)+(AP), (MSP) and (LUBP) are
equivalent.
1-6 limsup and liminf
第8周 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周 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周 11/17,11/19
Midterm examination
2-2 Open sets and interior of a set
第11周 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周 12/01,12/03
Chapter 3 Compact sets
3-1. Examples: the difference between I= [0,1] and I=(0,1]; consider continuou
s function on I
3-2 Sequentially compact: bisection process and bounded sequence; Heine-Borel
Theorem
3-3 Open cover and compact:
- examples
第13周 12/08,12/10
- compact implies bounded and closed; counterexample
- totally bounded;
- Bolzano-Weierstrass Theorem
第14周 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周 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周 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周 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
第18周 1/12
Final Exam
(这些是 Ceiba 上写的）
Ω 私心推荐指数(以五分计) ★★★★★
老师：★★★★（有趣老师）
喜欢看鬼灭：-★★★★★（老师说国中生才看鬼灭）
整体：★★★★
η 上课用书(影印讲义或是指定教科书)
1. Jerrold E. Marsden and Michael J. Hoffman, Elementary Classical Analysis, 2
nd 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
μ 上课方式(投影片、团体讨论、老师教学风格)
都写黑板，老师超有趣，可以去查俊全语录，但常常迟到，可以睡晚一点XD 。其他就，
我觉得老师很神，上课每次的证明好像都记在脑子里，好像没看过他带任何笔记，每次都
只有带咖啡#
σ 评分方式(给分甜吗？是扎实分？)
1. homework and quiz 25%
2. midterm exam 35%
3. final exam 40%
我觉得是扎实分，期中平均78，期末平均49，但应该是有调分的。
ρ 考题型式、作业方式
考题我放在考试版了，会考上课证明跟作业，考前一周上的内容也是必考。BTW, 期中超
简单，但期末直接大暴死QQ，不知道是不是每届都这样，你各位自己注意啊。
作业大概 2/3 简单、1/3 难，每周大概会花4-5小时写作业，有时候更久。
ω 其它(是否注重出席率？如果为外系选修，需先有什么基础较好吗？老师个性？
加签习惯？严禁迟到等…)
一起写在总结OuO
Ψ 总结
我觉得是外系的要想清楚，自己为啥要修这门课吧，我当初是想了解一些，常在论文中看
到的名词跟概念。一学期下来虽然有学到，但其实我觉得有点不合时间成本，如果只是要
了解那些概念，自己去看书可能会比较快，但不可否认地，上课还是学到很多，算是有其
他意外的收获，例如：找 bound 的技巧、norm的一些等价概念......。总之，是有收获
的！