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課程名稱 |
實分析二
Real Analysis (Ⅱ) |
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開課學期 |
107-2 |
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授課對象 |
理學院 數學研究所 |
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授課教師 |
王振男 |
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課號 |
MATH7202 |
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課程識別碼 |
221 U2880 |
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班次 |
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學分 |
3.0 |
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全/半年 |
半年 |
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必/選修 |
必修 |
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上課時間 |
星期一3,4(10:20~12:10)星期三3,4(10:20~12:10) |
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上課地點 |
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備註 |
研究所基礎課。上課教室為天數304。
總人數上限：30人 |
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Ceiba 課程網頁 |
http://ceiba.ntu.edu.tw/1072MATH7202 |
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課程簡介影片 |
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核心能力關聯 |
核心能力與課程規劃關聯圖 |
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課程大綱
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為確保您我的權利,請尊重智慧財產權及不得非法影印
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課程概述 |
這是第二學期實分析課程，這學期主要內容實分析的一些應用。我們會先討論L^p空間，Banach and Hilbert spaces，簡單的富氏分析，函數的逼近。抽象測度及其積分，還有其他相關的測度例如Hausdorff measure。剩下時間可以介紹機率的基礎。 |
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課程目標 |
熟悉實分析的應用，抽象測度的建立。 |
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課程要求 |
作業，期中，期末考。 |
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預期每週課後學習時數 |
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Office Hours |
每週二 10:00~12:00 |
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指定閱讀 |
Measure and Integral. An Introduction to Real Analysis. Second Edition. Richard Wheeden and Antoni Zygmund, CRC Press, Taylor & Francis Group. |
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參考書目 |
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. |
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評量方式
(僅供參考) |
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No. |
項目 |
百分比 |
說明 |
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1. |
Homework |
40% |
Weekly homework assignments |
2. |
Midterm |
30% |
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3. |
Final |
30% |
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週次 |
日期 |
單元主題 |
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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. |
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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. |
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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. |
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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. |
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第5週 |
3/18,3/20 |
Measurable functions with a given $\sigma$-algebra, Definition of an integral, Properties of integrals. |
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第6週 |
3/25,3/27 |
Absolutely continuous set functions, Singular set functions, Characterizations of absolute continuity and singularity, Hahn decomposition, Lebesgue decomposition |
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第7週 |
4/01,4/03 |
Quiz session on 4/1 |
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第8週 |
4/08,4/10 |
Midterm on 4/8, Proof of Lebesgue decomposition theorem, Radon-Nikodym theorem, Mutually singular measures. |
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第9週 |
4/15,4/17 |
Dual space of L^p, Relative differentiation of measures, Bounded overlaps, Besicovitch covering lemma, Regular Borel measures. |
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第10週 |
4/22,4/24 |
Self study week |
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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. |
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第12週 |
5/06,5/08 |
Lebesgue-Stieltjes integrals and Riemann-Stieltjes integrals, Hausdorff measure, Metric outer measure, Hausdorff dimension. |
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第13週 |
5/13,5/15 |
Caratheodory-Hahn extension theorem, Algebra, Extension of a measure, Characterization of finite Borel measures in R |
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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. |
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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. |
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第16週 |
6/03,6/05 |
Quiz session on 6/3. Distributions with compact supports, Fourier transform, Schwartz space, Tempered distributions. |
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第17週 |
6/10,6/12 |
Final on 6/10 |
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