課程資訊

Real Analysis (Ⅱ)

104-2

MATH7202

221 U2880

Ceiba 課程網頁
http://ceiba.ntu.edu.tw/1042MATH7202_RA2

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

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

Real Analysis I

Office Hours

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

(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