Functional Analysis

Fall 2011, Block 1

Course Description: In this course we develop some of the fundamental tools of Banach space theory, including the Hahn-Banach Theorem, as well as duality theory, main results connected with the Baire category theory (the open mapping theorem, the closed graph theorem, and the uniform boundedness principle) and infinite dimensional convexity results (extreme points, Krein-Milman theorem). We also discuss the following topics:

• Weak topologies on Banach spaces (including the Banach-Alaoglu theorem).
• Radon measures and the Riesz representation theorem for positive linear functionals.
• Compact operators on Hilbert spaces and the spectral theorem for self-adjoint compact operators on a Hilbert space.
• Fourier transform on R^n.

Textbook: Our main reference will be Gerald B. Folland’s book Real analysis: Modern techniques and their applications, Second Edition, Pure and Applied Mathematics (New York). A Wiley-Interscience Publication, John Wiley and Sons, Inc., New York, 1999, ISBN 0471317160. We will also use (particularly when discussing topics on compact operators on Hilbert spaces, but not only) Robert J. Zimmer’s book Essential results of functional analysis, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1990, ISBN 9780226983387.

Lectures with Magdalena Musat (musat@math.ku.dk): Mondays 10-12 (Aud. 10), Tuesdays 15-16 (Aud. 7) and Fridays 8-10 (Aud. 8).

Exercise Sessions with Isak Wulff Mottelson (dfk860@alumni.ku.dk): Tuesdays 13-15 (Room A 100) and 16-17 (Room A 105).

Lecture Notes

Homework and Mandatory Assignments