Honors Analysis

**Instructor: **Terence Tao,
MS 5622, ph. 206-4844 (tao@math.ucla.edu) Note: I get a lot of spam e-mails these days, so
please mark your e-mails with "Math 131BH" or something similar in
the subject line, or send the e-mails from an UCLA address, otherwise it may
get lost in the spam.

**Lectures: **MWF

**Section: **Thu

**Office hours: **M 1-3

**TA: **Nick Crawford, MS 6160, crawford@math.ucla.edu

**TA Office hours: **Tu 2-3, Th 11-12

**Textbook: ***Principles of Mathematical Analysis*, Walter Rudin, Third edition, International Series in Pure and
Applied Mathematics, McGraw-Hill Inc. 1976. The topics to be covered can
be found at http://www.math.ucla.edu/~tao/131bh.1.03s/schedule.html.
We will use both the textbook and my own lecture notes; it is strongly
recommended that you read both concurrently with the course (and not just when
it is time to do homework!). The book may be a challenge to read at first
- it is written in a professional, no-nonsense style, designed to convey
information as succinctly and precisely as possible - however if you make a
serious effort to go through all the theorems, proofs, definitions and remarks,
and understand them as much as possible, then you will get a lot more out of
this course (And you will be better prepared to handle other math classes in the
future). Note also that while you will
be examined on the material in the notes, we may not cover all of the notes
during the class. Thus I do expect you
to supplement your attendance at lectures with the reading of the notes.

**Homework: **Homework will be due on Fridays (in the TA mailbox or
handed to the TA directly) and returned every Thursday in section meetings,
starting on Friday, Apr 11; there will be nine assignments. Each
homework will consist of about ten problems of varying difficulty, about
half of which will be from the textbook. The exact questions will be available
on the Web, and also handed out in lectures. Only three of the questions,
chosen at random, will actually be graded, however it is strongly recommended
that you attempt all the questions in the assignment. These questions are
mostly theoretical; this course is one of the most theory-intensive ones in the
undergraduate syllabus, and one of the main purposes of the course (besides
learning the principles of analysis) is to teach you how to perform rigorous
proofs. Also, many of the homework questions cover core material of the
course, so it is essential that you know how to do these problems. You may use
whatever resources you wish to do the homework, including calculators,
textbooks, friends, TAs, etc. However, you should not just be content
with copying down someone else's answer to a homework question; it is important
that you understand why that answer is correct, and how you would go about it
if you had to do the problem on your own. Indeed, if you cannot do this,
you will have severe problems with the midterms and finals, so **when
reviewing for those exams it is important to be sure that you can do most of
the homework problems without external assistance**.

Solutions to selected questions from each homework may be available on the Web after the due date. Late homework will not be accepted. The lowest score of your nine assignments will be automatically dropped

**Java quiz:** A Java-based multiple choice quiz is
available on the class web page. This applet is designed to test your
basic knowledge of concepts, both on course material and on more foundational
material (such as logic, functions, the real numbers, set theory, etc);
questions will be added to this quiz as the course progresses. This quiz
is anonymous, and has no effect on your grade. However, it should be a
useful tool for discovering any weak spots in your knowledge of course
material. Please let me know if there are any technical difficulties in
loading or running the applet.

**Examinations: **There are two mid-terms on the Fridays of Apr 25 and
May 23 at 12-12:50, and a final on Wednesday, June 11, 3 - 6 pm (exam code 05),
at a room to be announced. You may choose an optional
"nickname" to go on your exams; this nickname will then be used when
the exam scores are posted (otherwise, your scores will be anonymous).

**Grading: **The final grade is based on the homework (15%, with the
lowest homework score dropped), two midterms (20%) each, and the final
examination (40%). In addition, there is an additional 5% which will be
automatically assigned to either your first midterm,
second midterm, or final, whichever one is superior. For instance, if you
scored well on the second midterm but poorly on the first midterm and final,
then the second midterm would be worth 25%, the final 40%, and the first
midterm 20%.

No reweighting beyond this 5% will be considered,
except in cases of genuine hardship (for instance, if there was a medical
emergency on the day of one of the midterms). Finally, there is an additional 2% of bonus points available; see below.

If you cannot make one of the examinations, contact me as soon as possible, preferably one week in advance of the exam. Retroactive, or last-minute requests for a make-up, will most likely be denied.

**Bonus points:** It is possible for students to earn bonus points to
improve their grade. Each point adds 0.1% to the final grade, and you may
earn a maximum of 20 points over the duration of the course. Thus the
theoretical maximum score for the class is 102%. You can earn a bonus
point whenever you discover an error (or have a good suggestion to make) in any
of my printed notes, homework, exams, solutions, class web page material, java
applet questions, or anything I say in Virtual
Office Hours. You can also earn a bonus point by asking or answering
questions in Virtual
Office Hours, or by suggesting a question for the Java
Quiz; more information on the bonus point system can be found at http://www.math.ucla.edu/~tao/131bh.1.03s/bonus.html.

Bonus points will be tracked on the class web page. Any attempt to abuse the bonus point system may result in forfeiture of all bonus points by the abuser.

**Calculators and written materials:** The exams will be open book, open
notes. The emphasis on the finals will be on theory and proofs;
calculators are unlikely to be useful, but clarity of writing and knowledge of
all the basic concepts, definitions, and key theorems will be.

**Mathematical level: **This is an honors course, and moreover it is a
highly theoretical one. In this course you will be encouraged to think
about concepts deeply, be able to create examples to illustrate such concepts,
write down precise definitions of these concepts, and then prove various
properties concerning these concepts. These proofs will require you to be
able to write precise mathematical statements, communicate in clear English, arrange an argument in a logical order, and present pictures
or examples as necessary to illustrate your work. These skills are not
innate and do require some practice, however if you work on the homework
problems, understand the proofs in the book, and review the solutions to the
homework, you will get better at these.

**Prerequisites: **This course
requires knowledge of the material in Math 131AH. In particular, you should be comfortable with
the theoretical aspects of real numbers, convergence of sequences and series,
differentiation and integration, and basic manipulation of sets. We will briefly review these concepts as we
go along in the course. In your homework
you are free to use any results from Math 131AH (or from any other course).

**World-Wide Web: **You are encouraged to visit the web-page for this
section at

http://www.math.ucla.edu/~tao/131bh.1.03s

This page will contain all the official information for the course, the latest homework, lecture notes, handouts, Virtual Office Hours, solutions to previous homework, sample exams and quizzes, updates, and other pieces of information.