Prerequisites:
Basic real analysis (i.e. the equivalent of a first year
graduate course in real analysis), including point set topology.
Basic linear algebra.
Recommended Books:
There will be no textbook. The following books
contain part of what I plan to cover in the course:
1) John B. Conway, A Course in Functional Analysis,
Springer GTM 96, 2nd edition (January 1997).
2) Barry Simon and Stephen Reed, Functional Analysis,
Academic Press, 1997, 2nd edition.
3) Gert Pedersen, Analysis Now,
Springer Verlag, GTM 118, 1988 (revised edition).
Syllabus:
The course will start off with a discussion of the basic principles
of functional analysis such as the Hahn-Banach theorem, the open mapping
theorem and the uniform boundedness principle. These are fundamental
theorems which are used in various areas of mathematics such as applied
analysis, representation theory, operator theory, operator algebras and
non-commutative geometry.
In the next part of the course we will discuss basic Hilbert space techniques.
These techniques play an important role in applications as for instance in
approximation theory, quantum mechanics and in the theory of wavelets. We
will then prove elementary facts about bounded operators on Hilbert space,
including compact and Fredholm operators. If time permits, we will present
the spectral theorem for normal operators on Hilbert space, which leads to
the so-called Borel functional calculus. Depending on time and interests
of the participants, applications to unbounded operators and operator
algebras will be discussed.
It is strongly recommended that those who want to continue in the spring
semester 2004 with Guoliang Yu's course on noncommutative geometry take
this course.
Grading:
There will be no exams. The course grade will be based on
attendance, homework problems and a presentation during class
on a topic of your choice (relevant to the course).