Fall Semester 2004, Functional Analysis A
(Math 362A)



Instructor: Dietmar Bisch
Lecture: TuTh, 9:35am-10:50am, SC 1404
Office: SC 1405, (615) 322-1999
Office hours: TuTh 10:50am-11:30am
Mailbox: SC 1326


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) John B. Conway, A Course in Operator Theory, American Math. Soc. (1999).
3) Barry Simon and Stephen Reed, Functional Analysis, Academic Press, 1997, 2nd edition.
4) 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, the closed graph 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 noncommutative geometry. We will then present topological vector spaces including the separation version of the Hahn-Banach theorem, weak topologies and Alaoglu's theorem. If time permits we will add a brief chapter on the theory of distributions.

In the next part of the course we will discuss basic Hilbert space techniques. These techniques play an important role in many different areas of science as for instance in quantum mechanics, quantum information theory, operator algebras, approximation theory and in the theory of wavelets. We will 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 functional calculus, an important tool in operator theory and operator algebras. Depending on time and interests of the participants, applications to quantum mechanics, unbounded operators and operator algebras will be discussed.

This course is a prerequisite for the course on von Neumann algebras and II1 factors which I plan to teach in the spring semester 2005.

Grading: 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). There will be no exams.