1
课程详述
COURSE SPECIFICATION
以下课程信息可能根据实际授课需要或在课程检讨之后产生变动。如对课程有任何疑问，请
联系授课教师。
The course information as follows may be subject to change, either during the session because of unforeseen
circumstances, or following review of the course at the end of the session. Queries about the course should be
directed to the course instructor.
1.
课程名称 Course Title
泛函分析（研究生） Functional Analysis（PG）
2.
授课院系
Originating Department
数学系 Mathematics
3.
课程编号
Course Code
MAT7003
4.
课程学分 Credit Value
3
5.
课程类别
Course Type
专业选修课 Major Elective Courses
（请保留相应选项 Please only keep the relevant information）
6.
授课学期
Semester
秋季 Fall
7.
授课语言
Teaching Language
英文 English
（请保留相应选项 Please only keep the relevant information）
8.
授课教师、所属学系、联系方
式（如属团队授课，请列明其
他授课教师）
Instructor(s), Affiliation&
Contact
（For team teaching, please list
all instructors）
Raul Ures，教授
Raul Ures, Professor
9.
实验员/助教、所属学系、联系
方式
Tutor/TA(s), Contact
无 NA
（请保留相应选项 Please only keep the relevant information）
10.
选课人数限额(可不填)
Maximum Enrolment
（Optional）
授课方式
Delivery Method
习题/辅导/讨论
Tutorials
实验/实习
Lab/Practical
其它(请具体注明)
Other（Please specify）
总学时
Total
11.
学时数
Credit Hours
48
2
12.
先修课程、其它学习要求
Pre-requisites or Other
Academic Requirements
泛函分析（本科）（MA302）
Functional Analysis (MA302)
13.
后续课程、其它学习规划
Courses for which this course
is a pre-requisite
14.
其它要求修读本课程的学系
Cross-listing Dept.
教学大纲及教学日历 SYLLABUS
15.
教学目标 Course Objectives
本课程是本科泛函分析课程的继续与深入，着重介绍有重要应用价值的经典理论，为学生的其他研究生数学课程和相关的
科研工作打下基础。
This course is a continuation of the undergraduate course “Functional Analysis”. It emphasizes the classical theories that
have important applications, laying a foundation for other related graduate courses and research.
16.
预达学习成果 Learning Outcomes
本课程强调抽象理论和具体应用的结合。通过一学期的学习，学生应掌握 Hahn-Banach 定理，并了解在正线性泛函的扩展
的应用，以及在凸规划问题的应用。对弱拓扑的有基本的掌握。掌握紧算子和 Fredholm 算子相关内容，并了解在二阶椭圆算
子和非局部扩散算子上的应用。通过学习一般的谱定理，加深对线性代数中有限维谱定理的认识，通过学习有限维和无限维
谱定理的差异，进一步认识无限维空间。并通过对有界对称，正规，酉算子以及无界自伴算子的谱理论的学习，对谱理论有更
深入的认识。通过学习本课程，还会为一些后续学习做铺垫，如通过对谱定理尤其是对希尔伯特空间上谱定理的学习，为
学生在量子力学的数学描述打下了基础。
This course is a balanced mix of abstract theories and applications. Through one semester of study, students should be
skilled at the Hahn-Banach theorem and understand the application of extensions in positive linear functionals and the
application of convex programming problems. And have a basic grasp of weak topology. Students should grasp the
contents of compact operators and Fredholm operators, and understand the application of second-order elliptic
operators and non-local diffusion operators. By learning the general spectral theorem, the understanding of the finite-
dimensional spectral theorem in linear algebra is deepened, and the infinite dimensional space is further understood by
learning the difference between the finite-dimensional and infinite-dimensional spectral theorems. And through the study
of the spectral theory of bounded symmetry, normal, unitary operator and unbounded self-adjoint operator, student will
have a deeper understanding in spectral theory. Through the study of this course, it will also lay the foundation for some
follow-up learning, such as the study of the spectral theorem, especially the spectral theorem in Hilbert space, which lays
a foundation for the mathematical description of quantum mechanics.
17.
课程内容及教学日历 （如授课语言以英文为主，则课程内容介绍可以用英文；如团队教学或模块教学，教学日历须注明
主讲人）
Course Contents (in Parts/Chapters/Sections/Weeks. Please notify name of instructor for course section(s), if
this is a team teaching or module course.)
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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
2. Weak and weak * topologies
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
2.4.3. Existence of PDE via Galerkin’s method
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
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
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
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
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
18.
教材及其它参考资料 Textbook and Supplementary Readings
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1.Functional Analysis, by Peter Lax.
2.Perturbation Theory for Linear Operators, by T. Kato.
3.泛函分析讲义（上、下），张恭庆等编著
课程评估 ASSESSMENT
19.
评估形式
Type of
Assessment
评估时间
Time
占考试总成绩百分比
% of final
score
违纪处罚
Penalty
备注
Notes
出勤 Attendance
课堂表现
Class
Performance
小测验
Quiz
课程项目 Projects
平时作业
Assignments
30
期中考试
Mid-Term Test
30
期末考试
Final Exam
40
期末报告
Final
Presentation
其它（可根据需要
改写以上评估方
式）
Others (The
above may be
modified as
necessary)
20.
记分方式 GRADING SYSTEM
A. 十三级等级制 Letter Grading
B. 二级记分制（通过/不通过） Pass/Fail Grading
课程审批 REVIEW AND APPROVAL
21.
本课程设置已经过以下责任人/委员会审议通过
This Course has been approved by the following person or committee of authority