課程資訊

Real Analysis (Ⅱ)

107-2

MATH7202

221 U2880

3.0

Ceiba 課程網頁
http://ceiba.ntu.edu.tw/1072MATH7202

Office Hours

1. Real Analysis (4th Edition) 4th Edition, Halsey Royden and Patrick Fitzpatrick.
2. Real Analysis and Probability (Cambridge Studies in Advanced Mathematics). 2nd Edition, R.M. Dudley.
3. Real Analysis: Modern Techniques and Their Applications. 2nd Edition, Gerald B. Folland.
4. Measure Theory and Fine Properties of Functions (Studies in Advanced Mathematics). 1st Edition, Lawrence C. Evans and Ronald F. Gariepy.

Measure and Integral. An Introduction to Real Analysis. Second Edition. Richard Wheeden and Antoni Zygmund, CRC Press, Taylor & Francis Group.

(僅供參考)

 No. 項目 百分比 說明 1. Homework 40% Weekly homework assignments 2. Midterm 30% 3. Final 30%

 課程進度
 週次 日期 單元主題 第1週 2/18,2/20 L^p spaces, Young's inequality, Holder's inequality, Minkowski's inequality, Minkowski inequality in integral form, l^p spaces, Banach spaces, Separability. 第2週 2/25,2/27 Continuity in L^p, L^2 spaces, Inner product spaces, Orthonormal system, Complete orthonormal system, Basis, Fourier coefficients, Bessel's inequality, Parseval's formula, Riesz-Fischer theorem, Isomorphism, Hilbert spaces, Weak convergence. 第3週 3/04,3/06 Convolution, Young's convolution theorem, Approximation of the identity, Denseness of L^p, Poisson Kernel, Gauss-Weierstrass kernel, Dirichlet problem, Heat equation. 第4週 3/11,3/13 Hardy-Littlewood maximal function, Mapping property of the Hardy-Littlewood maximal operator, The Marcinkiewicz integrals, Additive set functions, Continuity of the additive set function, Upper, lower, total variations, Jordan decomposition. 第5週 3/18,3/20 Measurable functions with a given $\sigma$-algebra, Definition of an integral, Properties of integrals. 第6週 3/25,3/27 Absolutely continuous set functions, Singular set functions, Characterizations of absolute continuity and singularity, Hahn decomposition, Lebesgue decomposition 第7週 4/01,4/03 Quiz session on 4/1 第8週 4/08,4/10 Midterm on 4/8, Proof of Lebesgue decomposition theorem, Radon-Nikodym theorem, Mutually singular measures. 第9週 4/15,4/17 Dual space of L^p, Relative differentiation of measures, Bounded overlaps, Besicovitch covering lemma, Regular Borel measures. 第10週 4/22,4/24 Self study week 第11週 4/29,5/01 Relative differentiation of measures, Outer measures, Caratheodory's characterization of a measurable set, Metric outer measure, Borel sets, Lebesgue-Stieltjes measures. 第12週 5/06,5/08 Lebesgue-Stieltjes integrals and Riemann-Stieltjes integrals, Hausdorff measure, Metric outer measure, Hausdorff dimension. 第13週 5/13,5/15 Caratheodory-Hahn extension theorem, Algebra, Extension of a measure, Characterization of finite Borel measures in R 第14週 5/20,5/22 Theory of distributions, Test functions, Continuous linear functional, Semi-norm estimate, Localization, Supports, Singular supports, Convergence in the sense of distribution. 第15週 5/27,5/29 Differentiations of distributions, Examples of distributions, Principal value of 1/x, Distributions constructed by analytic continuation, Primitive of a distribution, Distributions with compact supports. 第16週 6/03,6/05 Quiz session on 6/3. Distributions with compact supports, Fourier transform, Schwartz space, Tempered distributions. 第17週 6/10,6/12 Final on 6/10