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※ 本文是否可提供臺大同學轉作其他非營利用途？（須保留原作者 ID）
（是／否／其他條件）：
哪一學年度修課：
ψ 授課教師 (若為多人合授請寫開課教師，以方便收錄)
陳俊全
λ 開課系所與授課對象 (是否為必修或通識課 / 內容是否與某些背景相關)
數學系
δ 課程大概內容
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 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 f
orms a complete ordered field.
第6週 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 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
第18週 1/12
Final Exam
(這些是 Ceiba 上寫的）
Ω 私心推薦指數(以五分計) ★★★★★
老師：★★★★（有趣老師）
喜歡看鬼滅：-★★★★★（老師說國中生才看鬼滅）
整體：★★★★
η 上課用書(影印講義或是指定教科書)
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 上課方式(投影片、團體討論、老師教學風格)
都寫黑板，老師超有趣，可以去查俊全語錄，但常常遲到，可以睡晚一點XD 。其他就，
我覺得老師很神，上課每次的證明好像都記在腦子裡，好像沒看過他帶任何筆記，每次都
只有帶咖啡#
σ 評分方式(給分甜嗎？是紮實分？)
1. homework and quiz 25%
2. midterm exam 35%
3. final exam 40%
我覺得是扎實分，期中平均78，期末平均49，但應該是有調分的。
ρ 考題型式、作業方式
考題我放在考試版了，會考上課證明跟作業，考前一週上的內容也是必考。BTW, 期中超
簡單，但期末直接大暴死QQ，不知道是不是每屆都這樣，你各位自己注意啊。
作業大概 2/3 簡單、1/3 難，每週大概會花4-5小時寫作業，有時候更久。
ω 其它(是否注重出席率？如果為外系選修，需先有什麼基礎較好嗎？老師個性？
加簽習慣？嚴禁遲到等…)
一起寫在總結OuO
Ψ 總結
我覺得是外系的要想清楚，自己為啥要修這門課吧，我當初是想了解一些，常在論文中看
到的名詞跟概念。一學期下來雖然有學到，但其實我覺得有點不合時間成本，如果只是要
了解那些概念，自己去看書可能會比較快，但不可否認地，上課還是學到很多，算是有其
他意外的收穫，例如：找 bound 的技巧、norm的一些等價概念......。總之，是有收穫
的！