課程資訊

Real Analysis (Ⅱ)

108-2

MATH7202

221 U2880

3.0

Ceiba 課程網頁
http://ceiba.ntu.edu.tw/1082MATH7202_RA_II

1. Differentiation : Hardy-Littlewood maximal function, Lebesgue differentiation theorem, functions of bounded variation, absolutely continuous functions, differentiability of functions
2.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, Banach spaces, dual spaces
3. L^p spaces
4. Signed Measures: absolute continuity, Radon-Nikodym Theorem
5. Fourier Transform
6. Hausdorff Measure and Fractals

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

Course prerequisite: Introduction to Mathematical Analysis I, II; Real Analysis I

Office Hours

[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

Fon-Che Liu, Real Analysis, Oxford University Press

(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3週 3/02,3/04 1. Introduction 2. Maximal functions and Vitali covering lemma 第4週 3/09,3/11 1. Lebesgue differentiation Theorem 2. Lebesgue set of a function 3. Differentiation Theorem for a regularly-shrinking family of sets 4. Points of density 5. Rectifiable curves 第5週 3/16,3/18 1. point of density 2. Functions of bounded variation (BV): total variation, positive variation, negative variation 3. A function of bounded variation is the difference of two increasing functions 第6週 3/23,3/25 1. Vitali covering and Vitali Covering Lemma 2. An increasing function on [a,b] is differentiable almost every 第7週 3/30,4/01 1. Differentiation of functions of bounded variation 2. Absolutely continuous and fundamental theorem of calculus 3. Curve length and absolutely continuous 第8週 4/06,4/08 Basic principles of linear analysis: 1. Metric space, normed vector space, Banach space, bounded linear transformation 2. Baire category theorem 第9週 4/13,4/15 1. Principle of Uniform Boundedness 2. Banach-Steinhaus Theorem 3. The inverse of I -T; log (I+T) 第10週 4/20,4/22 1. Nowhere differentiable continuous functions 2. Open mapping theorem 第11週 4/27,4/29 midterm examination 第12週 5/04,5/06 Closed graph theorem; examples and counterexamples 第13週 5/11,5/13 1. Axiom of choice, Zorn's lemma, Hausdorff maximality principle 2. Hahn-Banach Theorem 第14週 5/18,5/20 1. Complex Hahn-Banach Theorem 2. Inner product space, Hilbert space 3. Parallelogram identity, Schwarz's inequality, triangle inequality 4. L2 space and completeness 第15週 5/25,5/27 1. Space l ² 2. Orthogonal projection theorem 3. Riesz representation theorem 4. Lebesgue-Nikodym theorem 第16週 6/01,6/03 1. Orthonormal family, Gram-Schmidt process, Hermite functions 2. Bessel's inequaity, Riesz-Fischer Theorem 3. Orthonormal basis and separability 第17週 6/08,6/10 1. Fourier series, Dirichlet kernel, Fejer kernel 2. Lp space, Young's inequality, Holder inequality 第18週 6/15,6/17 1. Completeness of Lp 2. The space lp 3. Signed measure: positive and negative sets, Hahn decomposition