FUNCTIONAL ANALYSIS AND REPRESENTATION THEORY
INSTRUCTOR: Viorel Nitica, Professor of Mathematics
COURSE DESCRIPTION: Our main goal is to introduce as rigourously as possible the representation theory of SL(2, R).
We will assume as known introductory topics in analysis and topology: topological spaces, Euclidean topology, continuous functions, compact spaces, metric spaces, complete- ness, Heine-Borel theorem, Baire category. For standard references, see 1) and 2) below. If needed, some of these topics will be covered during the weekly seminar, as well as in the individual meetings with the groups.
We start with an introduction to Lebesgue integration theory: Lebesgue Monotone Convergence Theorem, Fatou’s Lemma, Lebesgue Dominated Convergence Theorem. As an application we investigate Lp spaces. We continue with standard topics in Banach spaces: definition and examples, linear operators, Banach-Steinhaus Theorem, Open Map- ping Theorem, Closed Graph Theorem, Hahn-Banach Theorem; standard topics in Hilbert spaces: definition and examples, orthogonality, spectral theorem, unbounded operators; standard topics in Fourier analysis: Fourier series, Fourier transform, distributions, PDE with constant coefficients.
For the rest of the course we will read through the book "Noncommutative Harmonic Analysis” by M. T. Taylor and try to understand classical representation theory of compact Lie groups and of SL(2, R).
1) Topology without tears, web resource,
2) W. Rudin, Principles of real analysis, Elsevier, 1998
3) W. Rudin, Real and complex analysis, McGraw-Hill, New York, 1966
4) W. Rudin, Functional analysis, McGraw-Hill, 1991
5) M.T. Taylor, Noncommutative harmonic analysis, Mathematical Surveys and Monographs,
Vol. 22, American Mathematical Society, Providence, 1986