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課程名稱 |
實分析二
Real Analysis (Ⅱ) |
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開課學期 |
108-2 |
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授課對象 |
理學院 數學研究所 |
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授課教師 |
陳俊全 |
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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 |
新402新402 |
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備註 |
研究所基礎課。
總人數上限：60人
外系人數限制：15人 |
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Ceiba 課程網頁 |
http://ceiba.ntu.edu.tw/1082MATH7202_RA_II |
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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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課程概述 |
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 |
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課程目標 |
This course aims to introduce basic theory and techniques of modern analysis. |
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課程要求 |
Course prerequisite: Introduction to Mathematical Analysis I, II; Real Analysis I |
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預期每週課後學習時數 |
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Office Hours |
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指定閱讀 |
Fon-Che Liu, Real Analysis, Oxford University Press |
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參考書目 |
[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 |
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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. |
mid-term exam |
30% |
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2. |
final exam |
40% |
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3. |
homework |
30% |
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週次 |
日期 |
單元主題 |
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第3週 |
3/02,3/04 |
1. Introduction
2. Maximal functions and Vitali covering lemma |
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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 |
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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 |
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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 |
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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 |
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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 |
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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) |
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第10週 |
4/20,4/22 |
1. Nowhere differentiable continuous functions
2. Open mapping theorem |
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第11週 |
4/27,4/29 |
midterm examination |
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第12週 |
5/04,5/06 |
Closed graph theorem; examples and counterexamples |
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第13週 |
5/11,5/13 |
1. Axiom of choice, Zorn's lemma, Hausdorff maximality principle
2. Hahn-Banach Theorem |
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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 |
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第15週 |
5/25,5/27 |
1. Space l ²
2. Orthogonal projection theorem
3. Riesz representation theorem
4. Lebesgue-Nikodym theorem |
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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 |
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第17週 |
6/08,6/10 |
1. Fourier series, Dirichlet kernel, Fejer kernel
2. Lp space, Young's inequality, Holder inequality |
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第18週 |
6/15,6/17 |
1. Completeness of Lp
2. The space lp
3. Signed measure: positive and negative sets, Hahn decomposition |
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