Fall Semester 2017, Functional Analysis A
(Math 7120, formerly Math 362A)



Instructor: Dietmar Bisch
Lecture: TuTh, 9:35am-10:50am, SC 1313
Office: SC 1405, (615) 322-1999
Office hours (in SC 1405): TuTh 10:50am-11:30am & by appointment
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, Graduate Studies in Mathematics, 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 harmonic analysis, PDE's, representation theory, operator theory, operator algebras and noncommutative geometry. We then plan to 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 and perhaps one on fixed point theorems.

In the next part of the course we will discuss basic Hilbert space techniques. These techniques play an important role in many areas of mathematics, physics and computer science as for instance quantum mechanics, quantum information theory, operator algebras, operator theory, approximation theory, Fourier analysis, data analysis, game theory, potential theory and 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 we might take a few detours to quantum mechanics, group representation theory, distributions, fixed point theorems and operator algebras throughout the course.

Grading: The course grade will be based on attendance, weekly homework problems and perhaps an in-class presentation on a topic related to the course (the latter will depend on enrollment). There will be no exams.