课程大纲

COURSE SYLLABUS

1.

课程代码/名称

Course Code/Title

MAT8026 高等泛函分析 Advanced Functional Analysis

2.

课程性质

Compulsory/Elective

Compulsory

3.

课程学分/学时

Course Credit/Hours

3/48

4.

授课语言

Teaching Language

English

5.

授课教师

Instructor(s)

Raul Ures, Professor；

6.

是否面向本科生开放

Open to undergraduates

or not

Yes

7.

先修要求

Pre-requisites

（ 如 面 向 本 科 生 开 放 ， 请 注 明 区 分 内 容 。 If the course is open to

undergraduates, please indicate the diff

MA301 实变函数 MA202 复变函数 MA302 泛函分析

MA301Theory of Functions of a Real Variable, MA202 Complex Analysis,

MA302 Functional Analysis. No differences between undergraduate and

graduate students.

8.

教学目标

Course Objectives

（如 面 向 本 科 生开 放 ， 请 注 明 区 分 内容 。 If the course is open to undergraduates, please indicate the

difference.）

This course is the continuation of the same-named undergraduate course. It focuses on the classical theory that

are important to applications, preparing the students for other related courses and research. No differences

between undergraduate and graduate students.

9.

教学方法

Teaching Methods

（如面向本科生 开放， 请注 明区分 内容。 If the course is open to undergraduates, please indicate the

difference.）

The course will be taught in the standard way (“chalk and board”, in-class discussion, homework, office

hours, closed-book tests). The course is a balanced mix of abstract theories and applications.

No differences between undergraduate and graduate students.

10.

教学内容

Course Contents

（如面向本科生开放，请注明区分内容。 If the course is open to undergraduates, please indicate the

difference.）

Section 1

Hahn-Banach Theorem

1.1. The extension theorem

1.2. Hyperplane separation of convex sets

1.3. Applications

1.3.1 Extension of positive linear functionals

1.3.2 Lagrange multipliers of convex programming problems

Section 2

Weak and weak * convergence

2.1 Weak convergence and weak compactness of unit ball in reflexive

Banach spaces

2.2 Weak* convergence and weak* sequential compactness—Helly’s

Theorem

2.3 Banach-Alaoglu Theorem

2.4 Applications

2.4.1 Approximation of the delta-function by continuous functions

2.4.2 Approximate quadrature

Existence of PDE via Galerkin’s method

Section 3

General spectral theory

3.1. Spectral radius and Gelfand’s theorem

3.2. Functional calculus, spectral mapping theorem

3.3. Spectral decomposition/separation theorem

3.4. Isolated eigenvalues

3.4.1. Algebraic multiplicity

3.4.2. Laurent expansion of the resolvent operator near isolated eigenvalue

3.4.3. Stability of a finite set of isolated eigenvalues under small operator

perturbation

3.5. Spectrum of the adjoint operator

3.6. The case of unbounded but closed operators

Section 4

Compact operators and Fredholm operators

4.1. Riesz-Schauder theory

4.2. Hilbert-Schmidt theorem, min-max characterization of eigenvalues

4.3. Positive compact operators: Krein-Rutman theorem (for the special

case of Banach space C(Q), where Q is a compact Hausdorff space)

4.4. Fredholm operators

4.4.1. Characterization of Fredholm operators, pseudoinverse

4.4.2. Fredholm index: index of product of two operators, constancy of

index under small or compact perturbation

4.4.3. Essential spectrum of a bounded operator, and its constancy under

compact perturbation

4.5. Applications

4.5.1. Second order elliptic operators

4.5.2. Non-local diffusion operators

4.5.3. Toeplitz operators

Section 5

5. Spectral theory of bounded symmetric, normal and unitary operators

5.1. The spectrum of symmetric operators

5.2. Functional calculus for symmetric operators

5.3. Spectral resolution of symmetric operators

5.4. Absolutely continuous, singular, and point spectra

5.5. The spectral representation of symmetric operators

5.6. Spectral resolution of normal operators

5.7. Spectral resolution of unitary operators

5.8. Examples

Section 6

Unbounded self-adjoint operators

6.1. Spectral resolution via Cayley transform

6.2. The extension of unbounded symmetric operators, deficiency indices

6.3. The Friedrichs extension

6.4. Examples

Section 7

Semigroups of operators

7.1. Strongly continuous one-parameter semigroups

7.2. The generation of semigroups: Hille-Yosida theorem

7.3. Exponential decay of semigroups

7.4 Examples: semigroups defined by parabolic equations, and by

nonlocal diffusion equations

Section 8

Section 9

Section 10

…………

11.

课程考核

Course Assessment

（

○

1

考核形式 Form of examination；

○

2

.分数构成 grading policy；

○

3

如面向本科生开放，请注明区分内容。

If the course is open to undergraduates, please indicate the difference.）

Homework 30%+ Mid-term Exam (closed-book) 30%+Final Exam (closed book) 40%

12.

教材及其它参考资料

Textbook and Supplementary Readings

1. Functional Analysis, by Peter Lax.

2. 泛函分析讲义（上、下），张恭庆等编著

Perturbation Theory for Linear Operators, by T. Kato.