Math 5051,
Fall 2011
Instructor John E. McCarthy
Class
MWF 10.00-11.00 in 199 Cupples I
Office
105 Cupples
I
Office Hours MF 11:00-12:00, W 9:00-10:00
Phone
935-6753
Text Real Analysis: Modern techniques and their applications, Gerald B. Folland
Exams There will be two exams in the course:
1) Exam 1 Take home, Monday, October 17
2) Exam 2 Final exam, on Monday, December
19, 10.30-12.30
Homework
There will be weekly homework sets during the semester, handed out on Wednesday in class and due the following Wednesday. Some problems are fairly routine, but many are quite challenging.
Prerequisites The
pre-requisites for the course are the courses 417 and 418, and 4111,
or the
equivalent.
In
particular, I shall assume you are comfortable with basic topology, to the
level of
understanding the difference between
"compact'' and "sequentially compact'' (and
that in a metric space they coincide); and basic real
analysis, to the level of being able
to prove that if a
sequence of continuous functions converges uniformly their limit
is continuous, but that if they merely converge pointwise, their limit need not be continuous.
I also assume you know enough set theory to be happy about
using Zorn's lemma.
In particular, I shall assume known the Prologue and Chapter 4, sections 1-6, of Folland.
Content
The course is mainly about integration. We shall develop the
theory rigorously, because this is
necessary to
understand when one can interchange limits and integrals, which is what
analysts
spend all their time doing.
This semester, I aim to cover most of Chapters 1,2,3,5 and 6 of Folland.
Basis for Grading
Homework will be 20% of your grade (one assignment dropped), the midterm 30%,
the final 50%.
Homework
Homework is the most important part of the course. Whilst talking to other
people about it is not dis-allowed, too often this
degenerates into one person solving the problem, and other people copying them
(often justified to themselves by saying "I provide the ideas, X does the
details" - but the details are the key. If you can't translate the idea
into a real proof, you don't understand the material well enough). So I shall
introduce the following rules:
(a) You can only talk to some-one else about a problem if you have made a genuine
effort to solve it yourself.
(b) You must write up the solutions on your own.
Note, too, that a significant part of this course is learning how to write mathematics. Thus serious attention will be paid to how the solutions are written up. They should be written in full English sentences, and it should be clear what is going on (so, for example, if the argument is lengthy, dividing the proof into well-marked stages is a good idea). Faulty logic will be penalized with negative points. So, if you cannot prove something is true in general, but can do so if you assume an extra hypothesis, you should say so, and state the extra hypothesis you need. This will earn partial credit. But if you claim to prove the result in general, whilst implicitly assuming this hypothesis, you will lose many points, and may end up getting less than zero on a problem. If you really have no idea how to prove something, don't waffle. If you have an idea that you cannot make into a rigorous proof, say "This is an idea, but I cannot make it into a rigorous proof".
I do expect you to come to class every day, and to participate in class discussions. I also expect you to stay abreast of the material we are covering, and may call on you at any time to answer a question.
Please feel free to talk to me about the course at any time. I shall not tell you how to do current homework problems, but if you've made some progress, I may give hints.
Bibliography
The following is a brief bibliography you may find useful. Most books on real analysis cover the same material, so choose one whose style suits you.
W.Rudin
Real
and complex analysis
H.Royden Real analysis
E. Lieb and M. Loss Analysis