Spring 2025
Instructor: Marc Rieffel
Lectures: MWF 12:10-1:00, Evans 740
Course Control Number: 26113
Office: 811 Evans
Office Hours: M 2:15--3:15; W 10:30--11:30; F 10:30--11:30
GSI: Qiuyu Ren
Office: 775 Evans
Office Hours: T 4:00--5:00; Th 5:00--6:00
Prerequisites:
Math 202A or equivalent. I have no restrictions on enrollment
by undergraduates.
But undergraduates must fill out a form that
can be found by going from the department home web page
to "courses", then "enrollment" then "enrollment guidelines".
Students who did not take Math 202A last Fall and want to enroll in this Math 202B should
have a solid understanding of the following parts of the Lang text listed below:
Chapter II, Section 3 of Chapter III, and Sections 1-7 of Chapter VI.
Recommended Texts (available free on-line):
Real and Functional Analysis 3rd ed. by Serge Lang, Springer-Verlag
Basic Real Analysis by Anthony Knapp, Birkhauser.
Advanced Real Analysis by Anthony Knapp, Birkhauser.
Analysis Now by Gert K. Pedersen, Springer-Verlag.
Measure Theory by Paul Halmos, Springer-Verlag (a classic).
Real Analysis for Graduate Students by Richard F. Bass.
Functional Analysis by Richard F. Bass.
General Topology by John L. Kelley (a classic).
General Topology by Nicolas Bourbaki (a classic. Read "Advice to the reader".)
Measure, Integration & Real Analysis by Sheldon Axler.
Real Analysis by Bruce Blackadar. This is a preliminary version of a
remarkable book-in-progress.
The Lang text gives a presentation
of the material that is somewhat closer to that which
I will give than do the other texts.
My understanding is that through an agreement
between UC and the publishers, the texts by
Lang, Knapp, Pedersen, Halmos and Bourbaki are available
for free download by UC students. Here is the link for the
Lang text,
and the
first Knapp text, and the
second Knapp text, and the
Pedersen text, and the
Halmos text, and the
Bourbaki text.
You may need to use campus computers to authenticate yourself
to gain access.
Syllabus:
This course, and Math 202A, are "tool courses", in that
they cover some basic mathematical concepts that are of importance in virtually all areas of
mathematics and its applications. Our Math 202B will follow on from where we left off at the end
of Math 202A. The topics we will discuss include:
Product measures and integrals, the Hahn-Banach Theorem, duals
of Banach spaces and weak topologies, the Krein-Milman Theorem, Hilbert spaces,
the Radon-Nikodym Theorem, the Stone-Weierstrass Theorem, signed
measures, Radon measures, operators on Banach and Hilbert spaces, additional topics as time allows.
Grading:
I plan to assign roughly-weekly problem sets.
Collectively they will count for 50% of the course grade.
Students are
strongly encouraged to discuss the problem sets and the course content
with each other, but each student should write up their own solutions,
reflecting their own understanding, to turn in.
Even more, if students collaborate in working out solutions,
or get specific help from others, they should explicitly acknowledge
this help in the written work they turn in.
This is general scholarly best practice. There is no penalty for acknowledging such
collaboration or help. Some of the problems on the problem sets may
have solutions in various books or papers or other sources. I strongly recommend that
students try to solve the problems without looking at such sources. But
if such sources are used, then again, scholarly best practice is to
cite such sources.
There is no penalty for doing this. There will be a final examination, on
Wednesday, May 14, from 3 to 6 pm,
which will count for
35% of the course grade. There will be a midterm exam,
at the regular class time,
on Monday, March 17.
It will count for 15% of the course grade.
There will be no early
or make-up final examination. Nor will a make-up midterm exam be
given; instead, if you tell me ahead of time that you must miss the
midterm exam, then the final exam will count for 50% of your course
grade. If you miss the midterm
exam but do not tell me ahead of time, then you will need to bring
me a very persuasive doctor's note or equivalent
in order to try to avoid a score of 0.
Accommodations:
Students who need special accommodation for
examinations should bring me the appropriate paperwork, and must
tell me between one and two weeks in advance of
each exam what accommodation they need for that exam,
so that I will have enough time to arrange it.
Problem sets:
We will probably use GradeScope for posting
the problem sets and for students to upload
their solutions.
Changes: The above procedures are subject to change.
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.
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.
Since the output from TEX is a PDF file that can be directly uploaded to GradeScope,
this would eliminate the need to scan pages of paper for uploading to GradeScope.
This page was last updated on 01/26/2025
For the five free on-line books, here are the links for the
Bass Real Analysis,
Bass Functional Analysis,
Kelley Topology,
Axler Measure Integration Real Analysis, and
Blackadar Real Analysis.
The link for ancient lecture notes of mine on measures and integration
can be found at the bottom of my home web page.
In my lectures I will
try to give careful presentations of the material, well-motivated with examples.
If you use Mac OS, you can find it at MacTex.
If you use Windows or Linux, go to
Latex-project.
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}".