課程資訊

Real Analysis (Ⅰ)

108-1

MATH7201

221 U2870

3.0

Ceiba 課程網頁
http://ceiba.ntu.edu.tw/1081MATH7201_RA_I

Real Analysis is indispensable for in-depth understanding and effective application of methods of modern analysis. The course will cover the following topics:
1.Measure Theory: outer measure, Caratheodory outer measures, n-dimensional Lebesgue measure
2.Integration Theory: measurable functions, Lebesgue integral, monotone convergence and Lebesgue dominated convergence theorem, Fubini’s theorem
3.Elements of Functional Analysis: Baire Category Theorem and its consequences, open mapping theorem and closed graph theorem, separation principles and Hahn-Banach theorem, Hilbert spaces
4. Differentiation and Integration: Hardy-Littlewood maximal function, Lebesgue differentiation theorem, functions of bounded variation, absolutely continuous functions, differentiability of functions

This course aims to introduce basic theory and techniques of modern analysis.

Course prerequisite: Introduction to Mathematical Analysis I, II

Office Hours

Fon-Che Liu, Real Analysis, Oxford University Press

[1] Elias M. Stein and Rami Shakarchi, Real Analysis
[2] Richard Wheeden and Antoni Zygmund, Measure and Integral: An Introduction to Real Analysis
[3] Elliott H. Leib and Michael Loss, Analysis

(僅供參考)

 No. 項目 百分比 說明 1. mid-term exam 30% 2. final exam 40% 3. homework 30%

 課程進度
 週次 日期 單元主題 第1週 9/09,9/11 Introduction: 1. Bad functions: Weierstrass' nowhere differentiable function; Peano's space-filling function 2. Limits of integrals and Fourier series Volume of rectangles; open sets and cubes; Jordan outer measure and Lebesgue outer measure; 第2週 9/16,9/18 Properties of Lebesgue outer measure: monotonicity, countable sub-additivity, etc. 第3週 9/23,9/25 Measurable sets: open sets; closed sets; unions, intersections, and complements of measurable sets 第4週 9/30,10/02 Measurable sets: additivity for disjoint sets; limit property; translation and dilation properties of Lebesgue measure 第5週 10/07,10/09 1. σ algebra, Borel sets, axiom of choice, non-measurable sets 2. Definition of measurable functions 第6週 10/14,10/16 1. Properties of measurable functions: sum and product, composition with a continuous function, sup/inf and limsup/liminf, lim 2. Approximations by simple and step functions, Egorov's Theorem 第7週 10/21,10/23 1. Lusin's Theorem; Littlewood's three principles 2. Integration theory: integration of simple functions; integration of bounded functions supported on finite-measure sets 第8週 10/28,10/30 1. Bounded convergence theorem 2. Integration of non-negative functions: linearity, additivity, monotonicity, and other basic properties 第9週 11/04,11/06 1. General case: integrable functions, Dominated Convergence Theorem 2. Midterm examination 第10週 11/11,11/13 1. Invariance properties of integrals: translation, reflection, dilation; absolute continuity of an integrable function 2. L^1 space, completeness of L^1 space 第11週 11/18,11/20 Fubini's Theorem 第12週 11/25,11/27 1. Tonelli's Theroem; Applications of Fubini's Theorem 2. Abstract measure space 第13週 12/02,12/04 1. Construction of outer measures 2. Caratheodory measurable sets 3. Equivalence of Caratheodory measurability and Lebesgue measurability in Euclidean spaces 第14週 12/09,12/11 1. Premeasure and extension theorem 2. Integration theory on an abstract measure space: measurable function, Egorov's theorem, Fatou's Lemma, Monotone Convergence Theorem, Dominated Convergence Theorem 第15週 12/16,12/18 1. Product measure 2. Completion of the product measure of two Lebesgue measures 第16週 12/23,12/25 General Tonelli's theorems and Fubini's theorems 第17週 12/30,1/01 Polar coordinates and the corresponding product measure