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課程名稱 |
分析一
Analysis(Honor Program)(Ⅰ) |
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開課學期 |
112-1 |
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授課對象 |
理學院 數學系 |
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授課教師 |
林偉傑 |
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課號 |
MATH5232 |
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課程識別碼 |
221EU6540 |
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班次 |
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學分 |
5.0 |
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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 |
星期二2,3,4(9:10~12:10)星期四3,4(10:20~12:10) |
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上課地點 |
天數102天數102 |
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備註 |
本課程以英語授課。此課程研究生選修不算學分。
限學士班學生 且 限學士班二年級以上
總人數上限：60人 |
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課程簡介影片 |
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核心能力關聯 |
本課程尚未建立核心能力關連 |
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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 |
Mathematical analysis is a fundamental course that aims to provide us with advanced theorems and prepare us to write rigorous mathematical proofs. The main contents of this course in the first semester will cover most of the materials in the textbook. The contents in the second semester will first cover some basic Lebesgue theory. After that, we will study some selected advanced analysis topics. |
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課程目標 |
Develop abstract and logical thinking
Write rigorous mathematical statements and proofs |
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課程要求 |
Elementary analysis |
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預期每週課後學習時數 |
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Office Hours |
每週四 15:30~16:30
每週二 14:10~15:10 |
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指定閱讀 |
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參考書目 |
Real Mathematical Analysis by Charles C. Pugh
The book can be downloaded for free using NTU network. |
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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 |
40% |
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2. |
Midterm |
30% |
10/24 |
3. |
Final |
30% |
12/19 |
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週次 |
日期 |
單元主題 |
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第1週 |
9/5, 9/7 |
9/5: Metric spaces, sequences, continuity, homeomorphism, open and closed sets, topology, topological spaces, closures and neighborhoods in metric spaces
9/7: Closures and interiors in topological spaces, neighborhoods, continuity in topological spaces, Hausdorff spaces, subspace topology |
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第2週 |
9/12, 9/14 |
9/12: Basis, product topology, box topology, completeness
9/14: Completion, construction of real numbers using Cauchy sequences, sequential compactness, compactness |
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第3週 |
9/19, 9/21 |
9/19: Equivalence of sequential compactness and compactness in metric spaces, Lebesgue number, continuous functions and compactness, finite product of compact sets, subsets and compactness
9/21: Finite intersection property, Zorn's lemma, Tychonoff's theorem |
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第4週 |
9/26, 9/28 |
9/26: Connectedness, path connectedness, accumulation points, condensation points
9/28: Accumulation points, perfect sets, Cantor set, total disconnectedness |
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第5週 |
10/3, 10/5 |
10/3: Denseness, nowhere denseness, measure zero, Riemann integral, Darboux integral, Riemann integrability implies Darboux integrability
10/5: Darboux integrability implies Riemann integrability, sandwich principle, Lebesgue's integrability criterion |
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第6週 |
10/12 |
10/12: Consequences of the Lebesgue integrability criterion, fundamental theorem of calculus, the Cantor function, uniform convergence revisisted |
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第7週 |
10/17, 10/19 |
10/17: Space of bounded functions, space of continuous functions, equicontinuity, the Arzela-Ascoli theorem, applications
10/19: Extension of Arzela-Ascoli, Weierstrass approximation theorem |
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第8週 |
10/24, 10/26 |
10/24: Midterm
10/26: No class |
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第9週 |
10/31, 11/2 |
10/31: Stone-Weierstrass theorem, Banach fixed point theorem, Picard's theorem
11/2: Proof of Picard's theorem, analytic functions, continuous but nowhere differentiable functions, Baire category theorem |
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第10週 |
11/7, 11/9 |
11/7: Proof of the Baire category theorem, a generic continuous function is nowhere differentiable, artificial neural network, universal approximation theorem
11/9: Multiple integrals, Fubini's theorem, volume multiplier formula |
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第11週 |
11/14, 11/16 |
11/14: Change of variables formula, differential forms
11/16: Wedge products, exterior derivatives, pushforward and pullback |
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第12週 |
11/21, 11/23 |
11/21: Stokes' theorem, Brouwer's fixed point theorem
11/23: Proof of Brouwer's fixed point theorem, outer measure |
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第13週 |
11/28, 11/30 |
11/28: Properties of outer measure, measurable sets, Lebesgue measure, properties of Lebesgue measure
11/30: \sigma-algebra, Borel \sigma-algebra, nonmeasurable sets, approximation by simple or step functions |
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第14週 |
12/5, 12/7 |
12/5: Littlewood's three principles, Egorov's theorem, Lusin's theorem, Lebesgue integrals for simple functions and bounded measurable functions, bounded convergence theorem, Riemann integrals and Lebesgue integrals
12/7: Lebesgue integrals, Fatou's lemma, monotone convergence theorem, absolute continuity, Lebesgue dominated convergence theorem, the space of integrable functions |
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第15週 |
12/12, 12/14 |
12/12: Riesz-Fischer theorem, denseness of simple functions, step functions and continuous functions, invariances and continuity of Lebesgue integrals, Fubini's theorem
12/14: Proof of Fubini's theorem, Tonelli's theorem, slices and products of measurable sets, convolution |
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第16週 |
12/19 |
12/19: Final exam |
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