**Math 301, Real Analysis,
Spring 2006**

Harriet Pollatsek, 400 Clapp,
X 2341, hpollats@mtholyoke.edu

This information and weekly
assignments also appear on the course web page

(see www.mtholyoke.edu/courses/).

**Text:** *Introduction
to Analysis* by Arthur Mattuck, Prentice Hall, 1999.

**Homework** will be due on Monday and (normally) returned on Wednesday. Rewrites are due the following Monday (attach
the original). The higher of the two
“grades” will be recorded. There will be
10 homework assignments, including some exercises to be read by me (HP) and
some by a student assistant (SA). Each assignment will be graded “check plus”,
“check” or “check minus”. Papers graded
“check minus” should definitely be rewritten.
Graded homework counts for up to 220 points total out of 1000 (HP 150,
SA 70). Working in groups is strongly
encouraged, but you should write up the homework for me on your own, so I can
give you feedback on your writing. Some
exercises will be suggested but not assigned to hand in; these are valuable
practice and helpful to your learning. They can be used for oral presentations (see
below). In addition, there will be
almost weekly “challenge problems,” read by me.
These are optional, but some success with challenge problems is expected
for course grades of A or A-. You have
two weeks to work on each challenge problem; they too can be rewritten, and
rewrites are due the next week, same rules as for homework for me.

Note that the text includes
three levels of tasks: “questions” at the end of each section (for which
answers are provided at the end of the chapter), “exercises” keyed to sections
(from which homework assignments are usually chosen), and “problems” (which are
more challenging). You should answer all
the questions when you do the reading.

**Exams**: There will be
two take-home midterm exams worth 170 points each. The first, on chapters 1-6, will be handed
out 2/27 and due 3/6; the second, on chapters 7-15, will be handed out 4/10 and
due 4/17. *Note: The Science Symposium is 4/10, so the 4/10 class (exam review)
will be scheduled earlier so we can go to the Symposium---TBA.* Rewrites of the midterms are possible, and
the two scores will be averaged. The
final, worth 220 points, will be cumulative.
It will be a take-home exam due at the end of the exam period (earlier
for seniors). No rewrites of the final
will be possible (obviously).

**Quizzes** on definitions and statements of theorems will be on Wednesdays. You’ll always know exactly what material each
quiz will be drawn from. There will be eleven
quizzes at 20 points each, and I’ll drop the lowest two, so the maximum quiz
total will be 180 points. Precise
language is especially crucial in analysis, and grading of quizzes will be
picky.

**Oral presentations**: Everyone will
be graded on two oral presentations, each of one exercise, one before spring
break and one after. The two presentations
are worth a total of 16 points.
Consultation on presentations, with me and/or with other students, is
encouraged. Reserve presentation
problems ahead of time with me.

**Explorations on dynamical systems**: During the
weeks when you are working on exams, we will use class time for three modules
from *Discovering Dynamical Systems* by
Johnson, Madden and Sahin. You will use
Maple for guided explorations, looking for patterns, and you will work in small
groups to prove conjectures. Active
participation in the three explorations will be worth a total of 24 points.

**Schedule** (Do reading before class -- except 1/30; Fridays will
be regular classes.)

2/1: ch 1.1-1.4; 2/3: app A.0,
A.1; 2/3: ch 1.5-1.5, App A.3, A.4 p. 411

2/6: ch 2 and app A, A.2; 2/8: ch 3

2/13: ch 4; 2/15: ch 5

2/20 ch 6.1-6.3; 2/22: ch 6.4-6.5;
2/24: review

2/27 & 3/1 & 3/3:
Module 1, Introduction to dynamical systems (no reading)

3/6: ch 7; 3/8: ch 8.1

3/13: (ch 9), ch10; 3/15: ch
11

3/27: ch 12; 3/29: ch 13

4/3: ch 14; 4/5: ch 15; *TBA*: review

4/10 & 4/12 & 4/14:
Module 2, Classifying fixed points (no reading)

4/17: (ch 16), 17; 4/19: ch 18

4/24 ch 19; 4/26: ch 20

5/1 & 4/3 & 5/5: Module
3, Introduction to symbolic dynamical systems

5/8: metric spaces, review

**Foundations**: We will
assume that we know everything about the ordinary algebra of real numbers,
including the order properties. We will
place special emphasis on the ** completeness property** of the real
numbers. These properties taken together
(algebraic, order, completeness) completely characterize the real numbers as a
unique system and provide the working basis we will need in order to create an
analytic superstructure of grand proportions.
A warning: don’t be too impatient
to arrive at novel theorems. We first
need to go over old ground and strengthen our grasp of fundamental ideas. (Sometimes we’ll also assume facts from
calculus to illustrate the ideas, but eventually our point of view will shift,
and we will carefully

**Apology**: This term is
used in the same sense as the title *A Mathematician’s Apology *by G. H.
Hardy, the greatest English mathematician of the first half of the twentieth
century. The passage below, from his
book *Pure Mathematics,* refers to
the “limit of the sum is the sum of the limits” theorem for convergent
sequences.

… the
argument may possibly appear to the reader to be merely a piece of useless
pedantry, or an attempt to manufacture difficulties out of what is really
obvious. We do not assert that such an
opinion is, in this case, entirely groundless.
The result really is very obvious: nor would any mathematician think it
worthwhile as a rule to state arguments for what is so obvious at such length.

But the
reader must remember that the theorem, obvious though it may be, is one of the
most fundamental and important theorems in all mathematics. It is one which every mathematician uses,
consciously or unconsciously, twenty times a day. The proof of such a theorem must be made
absolutely clear, explicit, and rigorous: no room must be left for any possible
apprehension or confusion. And this is
not all. The great majority of theorems
concerning limits are, as the reader will discover before long, far from being
so simple and so obvious as this one. In
this case the result obviously indicated by common sense was true. In more difficult cases common sense as often
indicates an untrue result as a true one:
sometimes it fails to give any indication at all. In such cases, vague general arguments are
worse than useless: they lead to
mistakes not only gross in themselves but entirely confusing in their consequences. And unless the reader is prepared to take the
trouble to try and understand the way in which rigorous methods apply to simple
and obvious cases, where their application is easy, [s]he will find when [s]he
comes to difficult questions which cannot be settled without them, [s]he has
not the capacity to use them.