國立清華大學數學系
泛函分析一
Math 5030-00 (Fall 2014) Functional Analysis I
|
Office: 綜三 609 |
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.