# Math 571/572 (Functional Analysis). Spring 2008

Professor: Steve Krone

Office: 416 Brink Hall. Phone: 885-6317
Office Hours:

Class Time: MWF 10:30-11:20

Prerequisite: Math 535 (Measure Theory). It is possible to get around this prerequisite if you are comfortable with advanced calculus and are willing to do a little reading before and during the semester. See me about this.

Text: Lieb and Loss, Analysis, 2nd ed., AMS

Functional analysis arose from attempts to solve and analyze equations: differential equations, integral equations, matrix equations. It plays an essential role in the modern development of differential equations, probability theory, etc. Basically, one starts with an operator (for example, a second-order differential operator) and seeks information about functions that, when acted upon by the operator, satisfy an equation. A key part of the analysis involves deciding which "function space" is most appropriate. For example, if we are considering a differential operator, a natural first guess might be a class of suitably differentiable functions. However, this could restrict the behavior of solutions to such an extent that naturally occurring solutions would be eliminated; this might lead us to expand our function space to include "weakly differentiable" functions satisfying certain integrability properties (Sobolev spaces). In addition to situations like this in which functional analysis is actually used to analyze equations that arise in science and engineering, many of these ideas have crystalized into a beautiful abstract theory that is of interest in its own right (e.g., the general theory of Banach spaces and Hilbert spaces).

This is a 2-semester course in functional analysis. It will feature some of the "practical" applications alluded to above, as well as the classical "big results" of the general theory (Hahn-Banach theorem, Banach-Steinhaus uniform boundedness principle, spectral theory, etc.). The main text for the course (Lieb and Loss) is beautifully written and highly innovative. It is written by two people who actually use functional analysis, and their goal is to get you quickly up to speed on some of the key concepts and the underlying philosophy of the subject. It does not follow the tradition of many mathematics books that plod through the most general versions of all the theorems and never get to the "good stuff." I will supplement this with some of the more classical material from some of the texts listed below, especially in the second semester, which focuses mostly on spectral theory.

Outline of Topics:

571

• The main ideas and quick review of measure theory and integration
• Banach spaces and bounded linear operators
• L^p spaces, bounded linear functionals, weak convergence
• L^p inequalities of Hanner and Clarkson
• characterizing dual spaces
• convolutions and approximating L^p functions by smooth functions (mollifiers)
• General Banach spaces: uniform boundedness principle, open mapping theorem, bounded inverse theorem, closed graph theorem, Hahn-Banach theorem
• weak derivatives and the theory of distributions
• Sobolev spaces
• applications in differential equations

572

• Fourier transforms
• Fourier characterization of Sobolev spaces
• Sobolev inequalities
• generators and semigroups
• Spectral Theory
• spectrum, resolvent set, resolvent operators
• finite-dimensional case
• bounded linear operators on Banach and Hilbert spaces
• spectral mapping theorem and other tools
• spectral theory of compact operators
• spectral theory of bounded self-adjoint operators
• spectral theory of unbounded linear operators