## Math 206 Functional Analysis

### Fall 2023

Instructor: Marc A. Rieffel

Lectures: MWF 3:10-4:00 PM, in room 4 Evans

Course Control Number: 24537

Office: 811 Evans
e-mail: rieffel@math.berkeley.edu

Office Hours: M 1:45-2:45; W 12:45-1:45; F 1:45-2:45

Prerequisites: Math 202AB or equivalent is more than sufficient. Self-study of chapters 1 and 2 of the book by G. K. Pedersen mentioned below, far enough to understand the statements of the two main theorems in section 2.5, together with some understanding of the beginnings of the general theory of measure and integration, should be sufficient.

Recommended Text: The most comprehensive text is A Course in Functional Analysis, John B. Conway, 2nd ed., Springer-Verlag.
Also useful are Real and Functional Analysis, Serge Lang, 3rd ed., Springer-Verlag,
and Analysis Now, Gert K. Pedersen, Springer-Verlag,
in part because they also contain the material from Math 202AB that is needed for Math 206.
My understanding is that through an agreement between UC and Springer, the first edition of the Conway book, as well as the Lang and Pedersen books, are available for free download by students. You may need to use campus computers to authenticate yourself to gain access. You can find the Conway book here, while the Lang book can be found here, and the Pedersen book here. I do not plan to assign exercises from the Conway book, so the first edition should be satisfactory.

Description: Functional Analysis is concerned with vector spaces (often infinite-dimensional) of functions, often continuous, or differentiable, or measurable, and is also concerned with linear operators on these vector spaces. The key to being able to deal with the infinite-dimensionality of these vector spaces is that they are equipped with a topology, (often coming from a metric), so that one can discuss convergence and continuity on these vector spaces. The techniques that are developed in functional analysis find important use in most areas of mathematics and its applications.

Math 202AB includes a brief introduction to functional analysis. Math 206 continues development of this topic. We will study aspects of bounded operators, including compact'' and Fredholm'' operators, and develop the spectral theorem'' for self-adjoint operators on Hilbert spaces (possibly including for unbounded ones), and the related Commutative Gelfand-Naimark theorem. We will also develop the general theory of the Fourier transform and harmonic analysis. The theory of Banach algebras is a very elegant blend of algebra and topology which provides unifying principles for a number of different parts of mathematics and its applications, including the topics of Math 206, and we will develop and use the elementary part of this theory.

I will provide many interesting specific examples, and the problem sets will involve further important specific examples beyond those presented in class.

This course provides a strong foundation for Math 209, von Neumann Algebras, which is scheduled to be taught by Professor Voiculescu next Spring.

Grading: Many weeks I will give out a problem set, and the course grade will be based on the work done on these. There will be no final examination.

Using TEX: I encourage students to write up their problem-set solutions in TEX, more specifically LATEX. (But I do not require this.) LATEX is a powerful mathematical typesetting program which is widely used in the sciences, engineering, etc., for documents that use a lot of mathematical symbolism. Thus learning to use TEX is a valuable skill if you work in such fields.

Several guides to using TEX are listed on the department's computer support web pages. Others can be found by searching on-line for "latex tutorial'' or "latex manual''.

The best way to start learning TEX is not by trying to compose the long header that is needed, but rather by having a file that already has a header, and then gradually modifying that file as you learn how TEX works.
To obtain such a TEX file with a header, click
LATEX-sample . Make a copy to play with, once you have downloaded TEX onto your computer. At first, don't modify anything above "\begin{document}".

(most recent update: 8/24/2023)