若是通识课程评价，请用 [通识] 分类，勿使用 [评价] 分类 标题范例：[通识] A58 普通心理学丙 林以正 (看完後请用ctrl+y删除这两行) ※ 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？是紮实分？) 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，需先有什麽基础较好吗？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！ --

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