課程資訊

Real Analysis (Ⅰ)

104-1

MATH7201

221 U2870

Ceiba 課程網頁
http://ceiba.ntu.edu.tw/1041MATH7201_RA

The course will cover Chapters 1-4 of [1] and some contents of [2], including
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 theorm, Hilbert spaces
4.Lp-spaces
5.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

Textbooks:
[1] Elias M. Stein and Rami Shakarchi, Real Analysis
[2] Fon-Che Liu, Lecture notes in Real Analysis
Reference books:
[3] Richard Wheeden and Antoni Zygmund, Measure and Integral: An Introduction to Real Analysis
[4] Elliott H. Leib and Michael Loss, Analysis

(僅供參考)

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

 課程進度
 週次 日期 單元主題 第1週 9/14,9/16 Motivations for the Lebesgue theory: Limits of Riemann integrals; Fourier series; Measure=length, area, volume Axioms of measure Background (1870-1910): Weierstrass's nowhere differential function, Peano's space-filling curve 第2週 9/21,9/23 Lemma for rectangles and cubes; Outer measure (exterior measure); Outer Jordan content; "closed cubes" can be replaced by "open cubes" or "rectangles" in the definition of outer measure; Outer measure of a cube/rectangle; Properties of outer measure: outer measure of empty set, monotonicity, countable sub-additivity, sets separated by a positive distance, set can be written as countable union of almost disjoint cubes 第3週 9/28,9/30 Properties of open sets; Lebesgue measurable sets. 第4週 10/05,10/07 Properties of measurable sets Countable Additivity 第5週 10/12,10/14 sigma algebra; Borel sets; Non-measurable set; Invariance properties of measurable sets: translation, rotation, reflection and linear transformation 第6週 10/19,10/21 Measurable functions Approximation by simple functions and step functions 第7週 10/26,10/28 Egorove's theorem Lusin's theorem Convergence in measure 第8週 11/02,11/04 Integration theory Lebesgue integral: simple functions, bounded functions, nonnegative functions, general case Bounded convergence theorem Fatou's lemma 第9週 11/09,11/11 Bounded convergence theorem, Monotone convergence theorem, Fatou's lemma, Dominated convergence theorem 第10週 11/16,11/18 L1 space, completeness of L1 space; Linear transformation and integral; L1 continuity under translation 第11週 11/23,11/25 Fubini's theorem and Tonelli's theorem 第12週 11/30,12/02 Hardy-Littlewood maximal function, simple Vitali lemma; Lebesgue differentiation theorem 第13週 12/07,12/09 Points of Lebesgue density, Lebesgue points of a function, differentiation on sets shrinking regularly; Function of bounded variation 第14週 12/14,12/16 Good kernels and approximations to the identity 第15週 12/21,12/23 Simple Vitali lemma II; Vitali covering lemma 第16週 12/28,12/30 Differentiability of monotone functions and BV functions, Absolutely continuous functions 第17週 1/04,1/06 Lp space: Lp and L-infinite norm Young's inequality Holder and Minkowski inequalities Completeness of Lp space