課程資訊

Analysis(Honor Program)(Ⅰ)

110-1

MATH5232

221 U6540

5.0

Ceiba 課程網頁
http://ceiba.ntu.edu.tw/1101MATH5232_

Contents：This course serves as an introductory course for rigorous “Analysis”. It is a core subject for math-major students or those who are interested in rigorous and theoretical side of limits and calculus. The only prerequisite is the basic knowledge on calculus.

The materials in the first semester will include:
﹡Basic point-set topology
﹡Limits and Continuity
﹡Differentiability and functions of bounded variation
﹡Riemann-Stieltjes integral
﹡Infinite series and sequence of functions

At the end of the semester, students should have basic knowledge on the point set topology and are familiar with different concepts of continuity, differentiability, and convergence.

Office Hours

*Walter Rudin, Principles of Mathematical Analysis.
*William R. Wade, An Introduction to Analysis.
*Jerrold E. Marsden, Elementary Classical Analysis.

Textbook:
Tom M. Apostol, Mathematical Analysis.

(僅供參考)

 No. 項目 百分比 說明 1. 期中考 30% 2. 期末考 30% 3. 作業 20% 週作業 4. 小考 20% 週小考

 課程進度
 週次 日期 單元主題 第1週 Introduction and motivations. 第2週 Natural numbers and integers, Ordered set, Rationals, Upper and lower bounds, The least upper and the great lower bounds, The least upper bound property, Ordered field, Real number system, The completeness axiom. Denseness of rationals, The construction of the real number system, Dedekind cuts. 第3週 Countability, Cardinality, Metric and metric space, Discrete metric, Open balls, Open sets, Closed sets, Relatively open sets, Topology, Topological spaces, Component intervals, Characterization of an open set in R, Adherent points, Accumulation points. 第4週 Closure of a set, Characterization of a closed set by the adherent points, Derived set, Boundary points and boundary set, Bounded sets, Bolzano-Weierstrass theorem, Cantor intersection theorem, Open covering, Lindelof theorem, Second countability, Compactness. 第5週 Heine-Borel theorem, Equivalence of a compact set in R^n, Compactness and sequential compactness, Convergence, Cauchy sequence, Completeness, Total boundedness, Finite intersection property, Lebesgue covering lemma, Sequential compactness is equivalent to compactness in a metric space, Limits of a function, Continuity. 第6週 Continuous functions, Continuity and open sets, Preimage, Continuity and closed sets, Continuity and compact sets, Topological mappings (homeomorphism), Homeomorphic, Connectedness, Two-valued functions, Path-connectedness, Topologist's sine curve, Connected components, Characterization of open sets in R^n, Cantor set, Perfect set. 第7週 A perfect set is uncountable, Nowhere dense sets, Baire category theorem, Banach-Steinhaus theorem (principle of uniform boundedness), Uniform continuity, Contraction maps, Fixed-point theorem, Discontinuity. 第8週 Happy week. 第9週 11/16 (Midterm), Differentiation of a monotone function, Lebesgue's theorem, Vitali's covering lemma. 第10週 Proof of Vitali's covering lemma, Functions of bounded variation, Total variation, Characterization of a function of bounded variation, Rectifiable curves, Absolute continuity, Cantor-Lebesgue function, Riemann-Stieltjes integrals. 第11週 Integration by parts, Change of variable, Increasing integrators, Upper and lower Riemann sums, Upper and lower Riemann-Stieltjes integrals, Riemann's condition, Step-function integrators, Euler's summation formula, Integrators of bounded variation, Sufficient conditions for existence of Riemann-Stieltjes integrals. 第12週 Lebesgue's criterion for Riemann integrability, Mean-value theorems for Riemann-Stieltjes integrals, Fundamentals theorems of calculus, Integrals depending on a parameter, Differentiation under the integral sign, Exchange the order of integration. 第13週 Infinite series, Inserting and removing parentheses, Cauchy products, Cesaro summable, Infinite products, Cauchy's criterion, Euler's product for the Riemann zeta function, Sequence of functions, Pointwise and uniform convergence, Cauchy's criterion, Weierstrass M-test, Uniform convergence and continuity, Uniform convergence and Riemann integrability. 第14週 Uniform convergence and differentiability, Convergence in the mean, Power series, Analytic functions, Bernstein's theorem, Binomial series, Abel's limit theorem, Tauber's first theorem. 第15週 Power series and analytic functions, Sufficient and necessary condition of (real) analyticity, Nowhere differentiable (finite derivatives) function, Uniform boundedness, Equicontinuity, Arzela-Ascoli's theorem, Weierstrass approximation. 第16週 The denseness of nowhere differentiable functions, Weierstrass theorem, Baire category theorem. 第17週 Final exam.