Math 202B - Section 1 - Introduction to Topology and Analysis

Spring 2020

Instructor: Marc Rieffel

Lectures: TTh 9:40-11:00 am, Cory 241.

Course Control Number: 20484

Office: 811 Evans

Office Hours: Tuesdays 11-12, 1:30-2:30; Thursdays 11-12

GSI: Mitchell Taylor

GSI Office: 1042 Evans

GSI office hours: Mondays and Wednesdays 12:00-1:30

Prerequisites: Math 202A or equivalent. I have no restrictions on enrollment by undergraduates. See Math department staff advisors for any needed enrollment codes. 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).
Measure, Integration & Real Analysis by Sheldon Axler
The Lang text gives a presentation of the material that is somewhat closer to that which I will give than the other texts.
My understanding is that through an agreement between UC and the publishers, chapters of the texts by Lang, Knapp, Pedersen and Halmos are available for free download by students. You can find the chapters of the Lang text here, and chapters of the Knapp texts here, and here, chapters of the Pedersen text here, and chapters of the Halmos text here. You may need to use campus computers to authenticate yourself to gain access.
Links for the four free on-line books are:
Bass Real Analysis, Bass Functional Analysis, and Kelley Topology
Axler Measure Integration 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.

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 202B. The topics we will discuss include: The Hahn-Banach Theorem, duals of Banach spaces and weak topologies, Krein-Milman Theorem, Hilbert spaces, the Radon-Nikodym Theorem, Stone-Weierstrass Theorem, signed measures, Radon measures, operators on Banach and Hilbert spaces, additional topics as time allows.
In my lectures I will try to give careful presentations of the material, well-motivated with examples.

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.
There will be a final examination, on Wednesday May 13, 11:30-2:30 PM, which will count for 35% of the course grade. There will be a midterm exam, on Tuesday, March 17, at the regular class time. which will count for 15% of the course grade. Here is a
sample midterm exam, for a 50 minutes class rather than our 80 minutes class. 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 doctor's note or equivalent in order to try to avoid a score of 0.

Comments: Students who need special acomodation for examinations should bring me the appropriate paperwork, and must tell me at least a week in advance of each exam what specific acomodation they need for that exam, so that I will have enough time to arrange it.

The above procedures are subject to change.

Homework assignments: They will be posted at Homework as they are assigned.

Using TEX: I encourage students to write up their problem-set solutions in TEX, more specifically LATEX. This 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, you can find it at

Several guides to using TEX are listed on the department's computer support web pages. One beginner's guide can be found at
A more comprehensive guide can be found at

The best way to start learning TEX is not by trying to compose the long header, but rather by having a file that already has a header, and then gradually modifying that file as you learn how TEX works.
For 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}".

This page was last updated on 01/23/2020