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 quick review of the basic principles
of functional analysis, that is 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 quantum information theory. I then plan to
cover (locally convex) topological vector spaces including the separation
version
of the Hahn-Banach theorem, weak topologies and Alaoglu's theorem.
If time permits, I will add a brief chapter on the theory of distributions
and perhaps one on fixed point theorems.
In the next part of the course I will discuss basic Hilbert space techniques.
These techniques are significant in many areas of mathematics,
physics and computer science such as quantum mechanics, quantum
information theory/quantum computing, measured group theory, (group)
representation theory, operator algebras, operator theory, harmonic
analysis, Fourier analysis, data analysis, game theory, potential theory,
signal processing and wavelets. I will show elementary facts about
bounded operators on Hilbert space and will discuss projections,
partial isometries, the adjoint of an operator, unitaries, polar
decomposition, compact and Fredholm operators. If time permits,
I will present the spectral theorem for normal operators on Hilbert space,
which leads to a powerful functional calculus for normal operators, an
important tool in applications.
Depending on time and interest of the course participants, I might take
a few detours into quantum physics, group representation theory,
approximation properties of groups (amenability, property (T) etc.),
the theory of distributions/weak solutions of PDEs, noncommutative
measure theory & noncommutative topology, and other topics.
Grading: The typical course grade will be an A. I will assign
homework, so you can practice the concepts you learn in class. I expect
you to attend class meetings, and you should hand in at least some
of the HW sets. I might ask everyone to give an in-class presentation
on a topic related to the course (this will depend on enrollment).
There will be no exams.