FUNCTIONAL ANALYSIS (MTH 3106)

Professor M.Shubin

Fall 1996

Textbook:    Essential Results of Functional Analysis, by Robert J. Zimmer

The University of Chicago Press, Chicago, 1990

Office:  460 Lake Hall.     Phone:  Ext.5676     E-mail: shubin@neu.edu

Class meets:   Tuesday and Thursday 7:15 - 8:45 p.m., 544 Nightingale Hall

Functional Analysis developed in 20th century from an idea to treat functions as points in an infinite-dimensional space. This idea allows a miraculously successful use of rich geometric intuition when dealing with functions. It proved to be extremely fruitful in applications to differential equations, harmonic analysis, ergodic theory, group representations, quantum mechanics, economics models.

The aim of the course is to provide an introduction to essential results of Functional Analysis and some of its applications. The main prerequisite is the theory of Lebesgue integration, which is necessary mainly to understand examples, but at some moments is used in the theory itself. However the main abstract facts can be understood independently. Proofs of some important basic theorems about Hilbert and Banach spaces (e.g. Hahn-Banach Theorem and Open Mapping Theorem) will be omitted to allow more time for applications of the abstract technique. The principal topics to be covered are:

1.

Basics on operators in Banach and Hilbert spaces and operator topologies.

2.

Convexity and fixed point theorems. Haar measure. Krein-Milman theorem on extreme points.

3.

Compact operators. Peter-Weyl theorem for compact groups.

4.

Spectral theory. Gelfand's theory of commutative $C^\ast$-algebras. Mean ergodic theorem.

5.

Fourier transforms and Sobolev embedding theorems.

6.

Distributions and elliptic operators.

7.

Mathematical scheme of quantum mechanics.

Home assignments will be given weekly and will be a basis for your grade.

For additional reading and to cover occasional gaps that exist in the textbook I recommend the book ``Methods of Modern Mathematical Physics. I. Functional Analysis" by Michael Reed and Barry Simon, Academic Press. There are also very many other good books on Functional Analysis.

Textbook:    Essential Results of Functional Analysis by R.Zimmer, The University of Chicago Press, 1990

This textbook has a reasonably good set of problems which are usually not difficult (with a very small number of exceptions) provided you went over the preceding material. So it would be a good idea to try to solve as many of the problems from the book as you can (e.g. all of them). You can ask me questions about any problems (not necessarily assigned to you as homework). But do not accumulate a backlog: if you do, it would be difficult to catch up.

Keep in mind a very important role of examples. The examples are as important as theorems (if not more). So you have to familiarize yourself with as many examples of spaces, operators etc. as you can. Solving problems from the book serves this purpose though you will be better off if you also know examples which are discussed in the text of the book (and not only in the problems).

Assignment 1: Ch.1, p. 35-37, no. 4, 5, 9, 11, 13, 15, 19, 20, 21, 22.

Solutions to Assignment 1

Assignment 2: Ch.2, p. 50-51, no. 2, 3, 4, 5, 6, 9, 10, 11, 12, 13.

Solutions to Assignment 2

Assignment 3: Ch.3, p. 67-68, no. 1, 4, 5, 7, 8, 10, 11, 12, 14, 18.

Solutions to Assignment 3

Assignment 4: Ch.4, p. 93-94, no. 1, 4, 6, 7, 8, 9, 12, 13, 15, 17.

Solutions to Assignment 4