**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.

You can freely download versions of TEX onto your computer.

If you use Mac OS, you can find it at MacTex.

If you use Windows or Linux, go to
Latex-project.

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)