國立清華大學數學系
泛函分析一
Math 5030-00 (Fall 2014)  Functional Analysis I


Instructor: 陳國璋 Kuo-Chang Chen
Hours and days: F3F4F6
Classroom:
綜三 631
Prerequisites:
Real Analysis

Office: 綜三 609
Phone Number: (03) 5715131 Ext: 33067
Office Hours: By appointments
Email: kchen@math.nthu.edu.tw

 

Course Description:

Functional analysis is a mathematical subject which encompasses a variety of topics. You can easily find many books for this subject but with quite different contents. In modern days, it is often referred to abstract analysis for certain topological vector spaces (Hilbert spaces, Banach spaces, locally convex spaces, etc.) and linear operators on them. It can be considered an extension of linear algebra, which focuses mostly on finite-dimensional spaces. Most typical and important examples of abstract spaces in the study of functional analysis include L^p space, space of continuously differentiable functions, Sobolev space, Hardy space, etc. In order to better motivate contents of this course, it is advised that you take a year-long real analysis before taking this course.

 

Functional analysis is a fundamental mathematical tool for many other subjects, such as partial differential equations, calculus of variations, harmonic analysis, ergodic theory, quantum mechanics, probability, game theory, among others. In the first semester of this course I will focus on fundamental concepts and theorems, and spend most of our time on Hilbert spaces and Banach spaces. Topics to be included are:

 

1. Some topological vector spaces

2. Linear operators and functionals

3. The Hahn-Banach theorem

4. Linear operators on Hilbert spaces

5. Some fundamental theorems of Banach spaces

6. Weak and weak* topologies

7. Some fundamental theorems of locally convex spaces (if time permits)

 

Many further topics, such as compact operators, spectral theory, Banach algebra, index theory, are also vital parts of this subject. We shall cover them as well as some applications in the second semester.

 

This course is intended for graduate students and advanced undergraduate students who are interested in analysis and with solid undergraduate-level mathematical training. Students will be assumed to understand Lebesgue’s theory of integration, point set topology, and have some knowledge about differential equations. Other than that, we will try to make this course as self-contained as possible.

 

Textbook:

P.D.Lax: Functional Analysis, Wiley, 2002.

 

Primary References:

1. J.B.Conway: A Course in Functional Analysis, Second edition, Springer-Verlag, 1990.

2. E.M.Stein & R.Shakarchi: Functional Analysis: Introduction to Further Topics in Analysis, Princeton University Press, 2011.

 

Grading:

Homework assignments: 70%

Final exam: 30%

 

Attendance:

Students are expected to attend every scheduled class. It is the student's responsibility to keep informed of any announcements, syllabus adjustments or policy changes made during scheduled classes.