課程大綱

課程資訊

課程名稱

實分析二
Real Analysis (Ⅱ)

開課學期

104-2

授課對象

理學院數學研究所

授課教師

陳俊全

課號

MATH7202

課程識別碼

221 U2880

班次

學分

全/半年

半年

必/選修

選修

上課時間

星期一1,2(8:10~10:00)星期三3,4(10:20~12:10)

上課地點

天數102天數102

備註

研究所基礎課。
總人數上限：60人

Ceiba 課程網頁

http://ceiba.ntu.edu.tw/1042MATH7202_RA2

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課程概述

In this semester, this course will cover the following contents.
1. More properties of L^p spaces.
2.Elements of Functional Analysis: Baire Category Theorem and its consequences, open mapping theorem and closed graph theorem, separation principles and Hahn-Banach theorm, Hilbert spaces
3. Abstract measure and integration theory: exterior measure, Caratheodory's theorem, extension theorem, integration on a measure space, product measure and Fubini's theorem, signed measure and absolutely continuity of measure.
4. Hausdorff measure

課程目標

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

課程要求

Real Analysis I

預期每週課後學習時數

Office Hours

另約時間

指定閱讀

待補

參考書目

Textbooks:
[1] Elias M. Stein and Rami Shakarchi, Real Analysis
[2] Fon-Che Liu, Lecture notes in Real Analysis
Reference books:
[3] Richard Wheeden and Antoni Zygmund, Measure and Integral: An Introduction to Real Analysis
[4] Elliott H. Leib and Michael Loss, Analysis

評量方式
(僅供參考)

No.	項目	百分比	說明
1.	homework and quiz	30%
2.	mid-term exam	30%
3.	final exam	40%

課程進度

週次	日期	單元主題
第1週	2/22,2/24	Baire Category Theorem, nowhere differentiable functions
第2週	2/29,3/02	Normed vector space, Banach space, Principle of Uniform Boundedness, Banach-Steinhaus Theorem
第3週	3/07,3/09	Completeness of L(X,Y) if Y is complete Open mapping theorem, Applications of open mapping theorem
第4週	3/14,3/16	Closed graph theorem lp space: Holder inequality, Minkowski inequality, completeness
第5週	3/21,3/23	Dual space Dual space of Lp: Riesz representation theorem
第6週	3/28,3/30	Inner product space and Hilber space Parallelogram identity, Schwarz inequality, Triangle inequality Complete orthonomal system, Bessel's inequality, Parseval's identity A separable infinite dim Hilbert space is isometric to l2 Trigonometric functions form a complete orthonormal system on [0,2Pi]
第7週	4/04,4/06	Complete orthonormal system Bessel's inequality Parserval's identity Isometry between a separable Hilbert space and L2
第8週	4/11,4/13	General measure space Outer measure
第9週	4/18,4/20	Caratheodory condition Construction of a measure space
第10週	4/25,4/27	Egorov's theorem Fatou's lemma Monotone convergence theorem Dominated convergence theorem
第11週	5/02,5/04	Monotone convergence theorem Dominatd convergence theorem General integration theory
第12週	5/09,5/11	Unique extension from a premeasure to a measure
第13週	5/16,5/18	Product measure
第14週	5/23,5/25	Fubini's theroem Polar coordinates
第15週	5/30,6/01	Signed measure Total variation of a signed measure Hahn decomposition and Jordan decomposition
第16週	6/06,6/08	Lp space Lebesgue-Radon-Nikodym theorem
第17週	6/13,6/15	Lebesgue-Radon-Nikodym theorem Dual space of Lp space