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課程名稱 |
實分析一
Real Analysis (Ⅰ) |
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開課學期 |
111-1 |
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授課對象 |
理學院 應用數學科學研究所 |
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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 |
3.0 |
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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 |
天數101天數101 |
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備註 |
總人數上限：65人
外系人數限制：20人 |
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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 |
The first semester of this course will cover the following topics :
1. Measure Theory: outer measure, Caratheodory outer measures, n-dimensional Lebesgue measure
2. Integration Theory: measurable functions, Lebesgue integral, monotone convergence and Lebesgue dominated convergence theorem,
Fubini’s theorem
3. 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
The topics in the second semester will include:
4. Differentiation and Integration: Hardy-Littlewood maximal function, Lebesgue differentiation theorem, functions of bounded variation,
absolutely continuous functions, differentiability of functions
5. L^p spaces
6. Abstract Measure and Signed Measures: absolute continuity, Radon-Nikodym Theorem
7. Convolution operators and Fourier Transform
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課程目標 |
Course Goal：This course aims to introduce basic theory and techniques of modern analysis. |
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課程要求 |
Introduction to Mathematical Analysis I, II; Linear Algebra
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預期每週課後學習時數 |
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Office Hours |
另約時間 備註： TA's email
吳悠:R10221008@ntu.edu.tw
閻天立:R09221014@ntu.edu.tw |
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指定閱讀 |
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參考書目 |
(1) Fon-Che Liu, Real Analysis, Oxford University Press (2) Elias M. Stein and Rami Shakarchi, Real 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. |
Homework |
30% |
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2. |
Midterm Exam |
30% |
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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 |
9/5-9/7 |
0. Introduction:
Background and motivation:
(0) continuous nowhere differentiable functions, functions of bounded
variation, Cantor's set, a space-filling curve, Borel's measurable
sets, Lebesgue's theory, non-measurable sets
(1) Length, area, volume
(2) Fourier series
(3) Limits of Integrals
1. Measure theory
1-1 Properties of area (length, volume)
1-2 Rectangles and open sets |
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第2週 |
9/12-9/14 |
1. Measure theory
1-3 Jordan exterior (outer) measure
1-4 Lebesgue exterior (outer) measure
Examples, Cantor set,
Equivalent definitions of the Lebesgue exterior measure
Properties of exterior measure: monotonicity, sub-additivity |
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第3週 |
9/19-9/21 |
1-4 Lebesgue exterior (outer) measure
Properties of exterior measure: monotonicity, sub-additivity,
open sets approximation, disjoint sets with a distance
1-5 Measurable sets and the Lebesgue measure
Motivation
Basic properties:
(1) Open sets, measure zero sets, and closed sets are measurable.
(2) A countable union of measurable sets is measurable. |
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第4週 |
9/26-9/28 |
1-5 Measurable sets and the Lebesgue measure
(3) The complement of a measurable is measurable
(4) A countable intersection of measurable sets is measurable.
Theorem: Countable additivity holds for disjoint measurable sets
Sigma-algebra |
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第5週 |
10/3-10/5 |
1-5 Measurable sets and the Lebesgue measure
- Increasing and decreasing sequences of sets
- sigama-algebra and Borel sets: G_delta sets and F_sigma sets, relation
between Borel sets and Lebesgue measurable sets
- Non-measurable sets |
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第6週 |
10/10-10/12 |
- Invariance properties: translation, reflection, dilation
- Caratheodory measurable |
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第7週 |
10/17-10/19 |
1-6 Measurable function
- Riemann integral and Lebesgue integral
- Definition: equivalent definitions
- Continuous functions are measurable
- sup, inf , limsup, liminf, and lim of measurable functions
- sum and product of measurable functions |
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第8週 |
10/24-10/26 |
- measure zero
- almost everywhere
1-7 Approximation by simple functions and step functions
- approximation by simple function |
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第9週 |
10/31-11/1 |
- approximation by step functions
- midterm examination |
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第10週 |
11/7-11/9 |
1-8 Littlewood's three principles
- Egorov's Theroem
- Lusin's Theorem
- Littlewood's three principles
Chapter 2 Integration Theorey
2-1 The Lebesgue integral
- 4 steps: simple function, bounded functions supported on a set of
finite measure, non-negative functions, general functions |
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第11週 |
11/14-11/16 |
- Step 1 simple functions: Independence of representation, linearity,
additivity, monotonicity, triangle inequality
- Step 2 bounded functions supported on a set of finite measure: limit of
the integrals of convergent simple functions |
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第12週 |
11/21-11/23 |
- Step 2 bounded functions supported on a set of finite measure:
linearity, additivity, monotonicity, triangle inequality, Bounded
Convergent Theorem, Riemann integrable implies Lebesgue
integrable
- Step 3 nonnegative functions: linearity, additivity, monotonicity, |
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第13週 |
11/28-11/30 |
- Step 3 Fatou's lemma, Monotone Convergence Theorem
- Step 4 general case: integrable functions, linearity, additivity,
monotonicity, Dominated Convergence Theorem, invariance
properties
2-2 L1 space: completeness of L1 |
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第14週 |
12/5-12/7 |
2-2 L1 space: convergence in L1 implies convergence of a subsequence almost everywhere; approximation by simple functions, step functions, or continuous functions of compact support; invariance properties; complex valued functions |
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第15週 |
12/12-12/14 |
2-3 Fubini's Theorem: statement and proof |
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