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課程名稱 |
實分析一
Real Analysis (Ⅰ) |
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開課學期 |
101-1 |
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授課對象 |
理學院 數學系 |
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授課教師 |
陳俊全 |
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課號 |
MATH7201 |
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課程識別碼 |
221 U2870 |
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班次 |
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學分 |
3 |
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全/半年 |
半年 |
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必/選修 |
選修 |
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上課時間 |
星期一1,2(8:10~10:00)星期三3,4(10:20~12:10) |
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上課地點 |
天數102天數102 |
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備註 |
研究所基礎課。
總人數上限：60人 |
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Ceiba 課程網頁 |
http://ceiba.ntu.edu.tw/1011RA |
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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 |
In the first semester, we will focus on measure theory and Lebesgue integration in Euclidean space. Chapters 1-4 of [1] and some contents of [2] will be covered. |
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課程目標 |
This course aims to present basic theory and techniques of modern analysis. |
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課程要求 |
The prerequisite of this course is advanced calculus. |
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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 |
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參考書目 |
[1]. Real Analysis, Elias M. Stein and Rami Shakarchi
[2]. Measure and Integral: An Introduction to Real Analysis, Richard Wheeden and Antoni Zygmund.
[3]. Analysis, Elliott H. Leib and Michael Loss.
[4]. Real and Complex Analysis, Walter Rudin.
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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 |
25% |
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2. |
mid-term exam |
35% |
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3. |
final exam |
40% |
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週次 |
日期 |
單元主題 |
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第1週 |
9/10,9/12 |
Introduction
Topology of Euclidean spaces
Lebesgue exterior measure |
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第2週 |
9/17,9/19 |
exterior Jordan content, properties of exterior measure |
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第3週 |
9/24,9/26 |
Lebesgue measurable set |
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第4週 |
10/01,10/03 |
properties of measurable set, countable additivity |
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第5週 |
10/08,10/10 |
sigma algebra, Borel set, non-measurable set |
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第6週 |
10/15,10/17 |
measurable functions |
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第7週 |
10/22,10/24 |
Approximation by simple functions and step functions, Egorov's Theorem, Lusin's Theorem,
Littlewood's three principles |
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第8週 |
10/29,10/31 |
Integration theory: simple functions, bound functions, non-negative functions, general functions. |
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第9週 |
11/05,11/07 |
Bounded Convergence Theorem, Monotone Convergence Theorem, Fatou's Lemma,
Dominated Convergence Theorem |
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第10週 |
11/12,11/14 |
L1 space,
completeness,
Approximation by continuous functions, |
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第11週 |
11/19,11/21 |
Riemann integrable implies Lebesgue integrable,
Fubini's theorem |
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第12週 |
11/26,11/28 |
Fubini's theorem |
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第13週 |
12/03,12/05 |
Applications of Fubini's theory,
Tonneli's theorem |
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第14週 |
12/10,12/12 |
Translation, scaling, reflection invariances of intergral,
Absolute continuity of intergral,
Other properties |
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第15週 |
12/17,12/19 |
Hardy-Littlewood maximal function,
covering lemma |
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第16週 |
12/24,12/26 |
Lebesque differentioation function |
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第17週 |
12/31,1/02 |
Good keernel and approximation to the identity |
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