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課程資訊

課程名稱	泛函分析 Functional Analysis
開課學期	105-2
授課對象	理學院數學研究所
授課教師	王振男
課號	MATH5216
課程識別碼	221 U3900
班次
學分	3.0
全/半年	半年
必/選修	選修
上課時間	星期一3,4(10:20~12:10)星期三6(13:20~14:10)
上課地點	天數302天數302
備註	總人數上限：40人
Ceiba 課程網頁	http://ceiba.ntu.edu.tw/1052MATH5216_
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
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課程概述	This course is an introduction to the functional analysis. Roughly speaking, it can be seen as an extension of linear algebra to infinite dimensions. There are two major topics -- function spaces and linear operators. The most important part is the interaction of operators and function spaces.
課程目標	The aim is to introduce students basic knowledge of functional analysis, the framework of function spaces, and the theory of linear operators. Advanced students in analysis get acquainted with the analysis in infinite dimensions.
課程要求	Prerequisite: Undergraduate course on analysis (or advanced calculus), Real analysis.
預期每週課後學習時數
Office Hours	另約時間
指定閱讀	Methods of Modern Mathematical Physics I. Functional Analysis, by Michael Reed and Barry Simon, Academic Press, 1980.
參考書目	1. Functional Analysis, by Peter Lax, John Wiley & Sons, Inc. 2002. 2. Functional Analysis, by Kosaku Yosida, Springer, 1980.
評量方式 (僅供參考)

課程進度

週次	日期	單元主題
第1週	2/20,2/22	Review of Lebesgue integrals, Convergence theorems, Fubini's and Tonelli's theorems, Inner product spaces, Normed linear spaces, Parallelogram law, Pythagorean theorem, Bessel's inequality, Projection theorem.
第2週	2/27,3/01	Orthonormal basis, Zorn's lemma, Any Hilbert space has an orthonormal basis, Gram-Schmidt method.
第3週	3/06,3/08	Separable metric spaces, Countable orthonormal basis, Continuous linear functionals, Dual spaces, Riesz representation theorem, Lax-Milgram theorem, Banach spaces.
第4週	3/13,3/15	Some examples of Banach spaces, Dual and double dual spaces, Reflexive spaces, Hanh-Banach theorems, Applications of the Hanh-Banach theorem.
第5週	3/20,3/22	Reflexive Banach spaces, Canonical map, Quotient spaces, The dual space of a quotient space, Baire category theorem, Nowhere dense sets, 1st category, 2nd category.
第6週	3/27,3/29	Principle of uniform boundedness, Weak boundedness and strong boundedness, The Banach-Steinhaus theorem, Open mapping theorem, Closed graph theorem, Topological vector spaces, Locally convex spaces, Seminorms, Separating points, Seminormed linear spaces.
第7週	4/03,4/05	4/3 (holiday), 4/5 (holiday)
第8週	4/10,4/12	Convex, balanced, absorbing sets, Mikowski functional, Seminorms, Equivalent seminorms, Family of directed seminorms, Continuity of operators between locally convex spaces, Dual space of a locally convex space, Convex sets, Separation and strict separation, Geometric form of the Hahn-Banach theorem.
第9週	4/17,4/19	Proof of geometric form of the Hahn-Banach theorem, Normable LCS, Bounded sets in LCS, Metrizable LCS, Countable local base, Frechet spaces.
第10週	4/24,4/26	Weak topologies, Weak* topologies, Weak and weak* convergences, Mazur's theorem, Compactness, Banach Aloaglu theorem, Tychonoff's theorem.
第11週	5/01,5/03	Bounded operators, Norm topologies, Strong operator topologies, Weak operator topologies, Adjoint of an operator.
第12週	5/08,5/10	Self-adjoint operators, Normal operators, Operator norm of a self-adjoint operator, Unitary operators, Kernel of the adjoint of an operator and the orthogonal complement of its range, Neumann series, Projection operators, Complementary subspaces.
第13週	5/15,5/17	Compact operators, Completely continuous operators, Schauder's theorem, Convergence of compact operators, Finite rank operators, Hilbert-Schmidt operators, Riesz theorems for a compact operator, Finite dimensionality of N(L), Closedness of R(L), Riesz number, Fredholm alternative.
第14週	5/22,5/24	Analyticity, Analytic Fredholm Theorem, Spectrum of a bounded operator, Point spectrum, Continuous spectrum, Singular spectrum, Spectral radius, Spectrum of a compact operator, Riesz-Schauder's Theorem.
第15週	5/29,5/31	Hilbert-Schmidt theorem, Singular values decomposition, Examples.
第16週	6/05,6/07	Unbounded operators, Domain of an operator, Densely defined, Graph of an operator, Closed operators, Closable, Closure of a closable operator, The adjoint of an operator, Symmetric operators, Self-adjoint operators, Essentially self-adjoint, Criteria of self-adjointness.
第17週	6/12,6/14	6/12 Final exam (Open books).