課程資訊
課程名稱
 分析二
Analysis(Honor Program)(Ⅱ) 
開課學期
 109-2 
授課對象
 理學院  數學系  
授課教師
 蔡忠潤 
課號
 MATH5229 
課程識別碼
 221 U6550 
班次
  
學分
 5.0 
全/半年
 半年 
必/選修
 選修 
上課時間
 星期二2,3,4(9:10~12:10)星期四3,4(10:20~12:10) 
上課地點
 天數101天數101 
備註
 此課程研究生選修不算學分。
限本系所學生(含輔系、雙修生)
總人數上限：30人 
Ceiba 課程網頁
 http://ceiba.ntu.edu.tw/1092A2 
課程簡介影片
 
核心能力關聯
 本課程尚未建立核心能力關連
課程大綱
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課程概述

This is a course of mathematical analysis which provides solid training for mathematics majored students and students who are really interested in mathematical analysis. It provides the completely rigorous training in fundamental contents of mathematical analysis. Students are supposed to have the ability of writing rigorous mathematical arguments.

In this semester, we aim to follow closely "Real Analysis" by Stein and Shakarchi:
0. some general topology. (ch.1 in "Topology and Geometry" by Bredon)
1. measure, integration, differentiation (some overlap with the last part in last semester)
2. Hilbert space.
3. abstract measure, integration
4. Hausdorff measure 

課程目標
 Develop abstract and logical thinking.
Write rigorous mathematical statements and proofs. 
課程要求
 This course is a rigorous introduction to mathematical analysis. This semester will cover most of the material in the textbook by Stein and Shakarchi.

Course prerequisite:
1. freshman calculus (for math major)
2. linear algebra (for math major)
3. Analysis I 
預期每週課後學習時數
  
Office Hours
 每週二 14:00~15:00 
指定閱讀
 Elias M. Stein and Rami Shakarchi,
Real analysis. Measure theory, integration, and Hilbert spaces. 
參考書目
 Glen E. Bredon,
Topology and geometry. 
評量方式
(僅供參考)
  
No.
項目
百分比
說明
1. 
 Homework 
30% 
 You have two jokers: the lowest two grades will be discarded. 
2. 
 Quiz 
5% 
 3/16 
3. 
 Midterm 
30% 
 4/20 
4. 
 Final 
35% 
 6/15 
 
課程進度
週次
日期
單元主題
第1週
 2/23,2/25  topological space, subspace, connectivity, separation axiom, compactness.
ref: Bredon §1.2 ~ §1.7 except §1.6 
第2週
 3/02,3/04  product topology, Urysohn metrization theorem, one-point compactification, locally compact.
ref: Bredon §1.8 ~ §1.11 
第3週
 3/09,3/11  paracompact, partition of unity, quotient topology.
ref: Bredon §1.12 ~ §1.13

Lebesgue measure.
ref: Stein §1.2, §1.3 
第4週
 3/16,3/18  measurable function, Brunn--Minkowski inequality, Lebesgue integral via simple functions.
ref: Stein §1.4, §1.5, §2.1

3/16 第一次小考 
第5週
 3/23,3/25  Lebesgue integral (continued), space of integrable functions, Fubini theorem, Fourier transform of integrable functions.
ref: Stein §2.2, §2.3, §2.4 
第6週
 3/30,4/01  Fourier inversion (continued), Hardy-Littlewood maximal function.
ref: Stein §2.4, §3.1

4/01 清明連假 
第7週
 4/06,4/08  4/06 清明連假

Lebesgue differentiation theorem, approximations to the identity.
ref: Stein §3.1, §3.2 
第8週
 4/13,4/15  BV function, rectifiable curve, Minkowski content.
ref: Stein §3.3, §3.4 
第9週
 4/20,4/22  4/20 Midterm

isoperimetric inequality, Hilbert space, orthonormal basis.
ref: Stein §3.4, §4.1, §4.2 
第10週
 4/27,4/29  自主學習週 Spring Break 
第11週
 5/04,5/06  Poisson kernel, Fourier series of L1 and L2 functions, closed subspace, orthogonal projection, Riesz representation theorem, adjoint transform.
ref: Stein §4.3, §4.4, §4.5 
第12週
 5/11,5/13  integral operator, Hilbert--Schmidt operator, compact operator, spectral theorem of compact, self-adjoint operator.
ref: Stein §4.5 
第13週
 5/18,5/20  abstract measure, outer measure, Caratheodory measurability, metric outer measure, premeasure, Caratheodory extension, integration.
ref: Stein §6.1, §6.2 
第14週
 5/25,5/27  product measure, Fubini theorem, Riemann--Stieltjes integral, signed measure, absolute continuity of measures.
ref: Stein §6.3, §6.4 
第15週
 6/01,6/03  Radon--Nikodym theorem, two applications in probability theory, mean ergodic theorem, maximal ergodic theorem.
ref: Stein §6.4, §6.5 
第16週
 6/08,6/10  pointwise ergodic theorem, ergodic measure-preserving transformation, unique ergodicity, mixing, Hausdorff measure, Hausdorff dimension, Holder continuity, Cantor set.
ref: Stein §6.5, §7.1, §7.2 
第17週
 6/15,6/17  6/15 Final