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課程名稱 |
分析二
Analysis(Honor Program)(Ⅱ) |
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開課學期 |
112-2 |
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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 |
MATH5229 |
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課程識別碼 |
221EU6550 |
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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 |
選修 |
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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。
限本系所學生(含輔系、雙修生)
總人數上限：30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. In this semester, we will cover differentiation theory with respect to Lebesgue integrals, functional analysis and Fourier analysis. If time allows, we will cover some further advanced topic(s) in analysis. |
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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 |
Analysis (I) |
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預期每週課後學習時數 |
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Office Hours |
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指定閱讀 |
Stein and Shakarchi, Real Analysis. |
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參考書目 |
Kreyszig, Introductory Functional Analysis with Applications
Lax, Functional Analysis.
Lang, Real and Functional Analysis.
Stein and Shakarchi, Functional Analysis.
Stein and Shakarchi, Fourier Analysis: An Introduction.
Katznelson, Introduction to Harmonic 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 |
35% |
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2. |
Exam |
30% |
Date to be determined |
3. |
Final presentation and report |
35% |
Each group will read a research level paper / topic that is not covered in class. |
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週次 |
日期 |
單元主題 |
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第1週 |
2/20, 2/22 |
2/20: Hardy-Littlewood maximal function, Vitali covering lemma, Lebesgue differentiation theorem, local integrability, Lebesgue density theorem, Lebesgue set, approximations to the identity
2/22: L^1 and almost everywhere convergence of an approximation to the identity, functions of bounded variation, rising sun lemma, monotone continuous functions are differentiable almost everywhere |
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第2週 |
2/27, 2/29 |
2/27: Proof of that monotone continuous functions are differentiable almost everywhere, absolute continuity, fundamental theorem of calculus, jump functions
2/29: Differentiability of jump functions, rectifiable curves, arc length reparametrization, \ell^p spaces, linear functionals and dual space |
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第3週 |
3/5, 3/7 |
3/5: Bounded linear functionals, topological dual, operator norm, dual space of \ell^p, gauge, Hahn-Banach theorem, one-step extension
3/7: Proof of Hahn-Banach, applications, dual points, bounded linear functionals on the space of continuous functions on [a, b], Riemann-Stieltjes integrals, Riesz representation theorem |
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第4週 |
3/12, 3/14 |
3/12: Proof of the Riesz representation theorem, reflexive spaces, bounded linear operators
3/14: Transpose, annihilators, Fredholm alternative in infinite dimensions, examples of linear operators, uniform boundedness principle, resonance point, Fourier series |
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第5週 |
3/19, 3/21 |
3/19: Pointwise convergence of Fourier series, open mapping theorem, Banach inverse mapping theorem, closed graph theorem
3/21: Spectrum, spectral radius formula, Hilbert spaces, best approximation |
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第6週 |
3/26, 3/28 |
3/26: Orthogonal projection, self-duality of Hilbert spaces, Riesz representation theorem, orthogonal complement, Bessel's inequality, complete orthonormal sets, Parseval's identity
3/28: Separable Hilbert spaces, Fourier series in L^2, adjoint operators, self-adjoint operators, compact operators, examples, compact self-adjoint operators |
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第7週 |
4/2 |
4/2: Spectral theorem for compact self-adjoint operators, Sturm-Liouville eigenvalue problem, Green's function |
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第8週 |
4/9, 4/11 |
4/9: Spectral decomposition of Sturm-Liouville differetial operator, weak convergence, weak sequential compactness
4/11: Best approximation in reflexive spaces, weak*-convergence, weak*-sequential compactness, topology induced by functionals, separation theorem |
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第9週 |
4/16, 4/18 |
4/16: Proof of separation theorem, weak and weak*-topologies, Banach-Alaoglu theorem, weak compactness and reflexivity, extreme points
4/18: Examples, Krein-Milman theorem, Fourier series, Cesaro summability, Fejer kernel, uniqueness theorem, decay rate of Fourier coefficients, trigonometric series which is not a Fourier series |
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第10週 |
4/23, 4/25 |
4/23: Hausdorff-Young inequality, Riesz-Thorin theorem, Weyl's equidistribution theorem, Fourier transform, Riemann-Lebesgue lemma, Fejer kernel |
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第11週 |
4/30, 5/2 |
4/30: Midterm exam
5/2: Space of continuous functions, maximal ideals, characters, Banach algebras, Banach *-algebras, C*-algebras, examples, complex analysis on Banach spaces, Gelfand-Mazur theorem, properties of ideals and characters |
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