Fall 2024
Instructor: Marc Rieffel
Lectures: MWF 10:10-11:00, Etcheverry 3108
Course Control Number: 22022
Office: 811 Evans
Office Hours: M 11:45-12:30; W 1:15-2:45; F 11:15-12:00
GSI: Tianrui Xu
Office: TBA
Office Hours: TBA
Prerequisites: Math 104 and considerable experience with other upper-division mathematics courses in dealing with quite abstract concepts and with constructing somewhat complicated proofs. Math 105, 110, 142 and 185 give especially useful preparation. 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".
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 202B, are "tool courses", in that
they cover some basic mathematical concepts that are of importance in
virtually all areas of mathematics and its applications. (For lack of time
we will not be able to present any of the applications, beyond a few hints.)
In Math 202A we will cover: Metric spaces and general topological spaces, compactness,
theorems of Tychonoff, Urysohn, Tietze, locally compact spaces;
an introduction to general measure spaces and integration of functions on
them, with Lebesgue measure on the real line as a key example;
Banach spaces of functions, and the very beginnings of functional analysis.
In Math 202B most of these topics will be developed further,
especially measure and integration, and functional analysis.
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.
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 09/03/2024
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.
There will be a final examination, on
Monday, December 16, from 8 to 11 am,
which will count for
35% of the course grade. There will be a midterm exam,
at the regular class time,
on Monday, October 21 .
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.
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}".