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MAT 357H: Real Analysis I

__ Professor__: Mary Pugh

__ Contact information__: mpugh@math. utoronto.ca

__ Class times__: MWF 12:10-1:00

__ Office hours__: Tu 5:00-6:00

__ Office location__: room 3141, Earth Sciences Centre,
22 Russell Street
(To find my
office, enter the building on the Northwest corner of Huron and
Russell. Get to the third floor. Walk as far south as you can.
Now walk as far west as you can.)

**Wanna send me email?**
If you're going to send me email, please include "mat357" in the
subject line. Otherwise,
my spam filter
will catch your email. (Note: do not include a space between "mat"
and "357", the subject line should contain a six-symbol string.)

**Homework Policy**
In working on your homework,
you can invoke anything from earlier in the book, both theorems and
exercises. But do make sure to include an opening clause on the lines
of, "By exercise xx on page nn," You don't need to reprove/prove
the thing you're invoking.

Starting on Wednesday March 2, homework is due
in class by 12:10. If you can't come to class, give it to a classmate
to hand in. If it arrives after 12:10 it will be counted as late and
your mark will be docked by five points. (To date, each problem has
been worth 5 points. By handing something in late you'll be losing
a full problem's worth of credit.)
You are *actively* encouraged
to work in groups on the homework.

Given Professor Greiner's method of assigning course marks, I have no
way of learning who you are and what you know. As a result,
if you were to
ask me to write a letter of recommendation on your
behalf or if you were to apply grad school and I
received an email from a colleague asking about you, then I would be
unable to provide any sort of information. And so, I encourage you to
choose your favorite homework assignment (past or future) in which you
did all the work on your own (no help from classmates, TAs, or profs)
and declare on your honor that this is the case. I would then read
this homework carefully and extract information from it for future
use. Whether or not you choose to do this will have no effect on your
course mark.

**Is there something in the notes that confuses you?** It might be a mistake on
my part. Check the current version of the notes on the web --- sometimes I
find mistakes and correct them. If the current version has the same point of
confusion, please email me and we can figure out what's going on.

Lecture Notes: Feb 21, 2005.

**HW 6:** Due in class before 12:10 on Weds March 2.

page 157: 4, 6, 9

section 12A: "Prove or disprove: if A is measurable and B agrees with
A up to a set of measure zero, then B is measurable." Agreeing up to a
set of measure zero means that the measure of A Delta B is zero where
A Delta B is defined in problem 1, page 165.

page 165: 3, 4, 5

Lecture Notes: Feb 23, 2005.

Lecture Notes: Feb 25, 2005.

Lecture Notes: Feb 28, 2005.

**HW 7:** Due in class before 12:10 on Weds March 9.

Let X be a finite dimensional real vector space. Let ||.|| and |||.||| be
two norms on X. Prove that these norms are equivalent, using the
definition on page 6 of the Feb 28 lecture notes.

Consider the space C^1(R), defined on page 2 of the Feb 28 lecture
notes. Consider the norms ||.||_{inf,0} and ||.||_{inf,1} defined as:
*||f||_{inf,0} := ||f||_{inf}* and
*||f||_{inf,1} := ||f||_{inf}+||f_x||_{inf}*.
(Note that "inf" means "infinity". And that if f(x) = 2 + 4 cos(9 x)
then ||f||_inf = 6, ||f_x||_inf = 36, hence
||f||_{inf,0} = 6 and
||f||_{inf,1} = 42.)
First prove that these norms are not equivalent by showing
directly that one can never find a constant C so that
the definition of norm equivalence on page 6 holds.
Now prove that these norms are not equivalent by proving that
(C^1(R),||.||_{inf,0}) is not complete while
(C^1(R),||.||_{inf,1}) is complete. (Make sure to explain why this
disparity then implies that no C can exist.)

The proof of Lemma 12.6 in the book is very sketchy. Present a full
proof. For example, make it clear how you used that B is bounded, why you had
to introduce a new open set A, and prove that g_0 really is continuous.

page 168: 2, 9, 12. Note that in problems 2 and 9 you're expected to assume
that f_n, f, g_n, and g are all in L^2 and the convergence is in L^2
as opposed to pointwise or some
other type of convergence. (I.e. ||f_n - f||_2 goes to zero.)

page 172: 2-6.

Lecture Notes: Mar 2, 2005.

Lecture Notes: Mar 4, 2005.

Lecture Notes: Mar 7, 2005.

**HW 8:** Due in class before 12:10 on Weds March 16.

Let X be a real inner product space. A *subspace* of X is a
subset of X that is closed under vector addition and scalar
multiplication. An orthonormal collection of vectors is an
*orthonormal basis* for X if X equals the closure of the subspace
spanned by the orthonormal collection. I.e. given x in X you can find
a sequence x_n such that each x_n is a (finite) linear combination of
vectors in the orthonormal collection and ||x-x_n|| -> 0 as n ->
infinity. Prove that if X is separable (has a countable dense subset)
then X has an orthonormal basis.

Consider the Hardy-Littlewood maximal function defined on page 169.
Prove or disprove: "If h is in L^1 then h*
is a measurable function."

In the proof of the covering lemma on pp 170-171, the author uses a
factor of 1/2, ultimately leading to the constant 5 in the statement
of the theorem. If one used a factor of lambda, where 0 < lambda < 1,
what would be the 5 be replaced by? (Present your reasoning; don't just
give an answer w/o explaining your logic.) This suggests that this
method of
proof can never result in a constant less than or equal
to what number?

Given an integrable function h, we define the function *A_delta h* by
*|Ix|*(A_delta h)(x) = int_{Ix} h*
where Ix is an interval of length 2*delta, centered at x. I.e., (A_delta h)(x)
is the average value of h, as averaged over the interval Ix.
Prove that (A_delta h)(x) is a continuous function of x.

page 171: 1-3.

Lecture Notes: Mar 9, 2005.

Lecture Notes: Mar 11, 2005.

Lecture Notes: Mar 14, 2005.

**HW 9:** Due in class before 12:10 on Weds March 23.

page 179: Problem 1: find the Fourier coefficients for the functions given in a, b, d, and for f(x) = |x|x. Discuss how the smoothness of f appears to be reflected in the large-k behaviour of the Fourier coefficients.

page 179: Problems 5, 6, 8, 9

page 184: 1, 2

The following theorem about convolutions is true: "If f is in
L^1(R) and g is in L^p(R) then f*g is in L^p(R) and ||f*g||_p <=
||f||_1 ||g||_p." (We can't discuss the proof because it depends on
Fubini's theorem, which is slightly beyond the reach of the course.)
Prove the following: "Assume g is in L^1(R). If f and f_x are in
C_o(R) then both f*g and (f*g)_x are continuous." Now prove: "If
f and its first k derivatives are in C_o(R) then f*g is continuous
and has k continuous derivatives."

If f and f_x are continuous then what additional assumption would
imply that if g is in L^1 then f*g and (f*g)_x are continuous? (Please
find an assumption that's weaker than having compact support.)

Prove or disprove: the above statements are true for g in L^p(R)
where 1 < p < infinity.

Here's an interesting theorem that you can't use in your HW: "If 1/p + 1/q = 1
and 1 <= p <= infinity and f is in L^p and g is in L^q then f*g is bounded
and uniformly continuous."

**Sad news: my note-taking computer has died. There will be no more
online notes this semester.**

Here is a *.ps file
of the graph that I'm handing out in class on Friday. And here
is the matlab
*.m file that I used to make the graph. In case you
have access to matlab.

**HW 10:** Due in class before 12:10 on Weds March 30.

page 189: Problem 2

page 191: Problem 2

page 193: Problems 1, 3, 5, 8, 9, 10

Here's a New York Times interview with one of my heroes, the wonderful
Peter Lax. Not only is he an amazing mathematician, but he's
a delightful, kind person. And he cares about communicating his
love of mathematics, both in person and through his excellent books.
*
Note: if you're not UofT affiliated (and hence wouldn't have access to this article through our e-resources) then you are violating copyright law if you look at this article.*

Here are the
*.ps
*files
of the graphs that I handed out in class on Wednesday, March 30. And here
is the matlab
*.m file that I used to make the graph. In case you
have access to matlab. I used matlab's fft and ifft packages. To see
how they work, check out the "How matlab does its fast Fourier transform"
page
here.

**Putnam Party!** There will be a talk and awards ceremony on
Tuesday April 12 at 3pm in SS5017A.
Check it out.

**HW 11:** Due in class before 12:10 on Fri April 8.

page 196: 3,4

page 199: 1, 2

page 214: 1, 7, 9. Please make sure that you can do problems
2-6, but don't hand them in. (They're simple but important exercises.)

page 217: 3, 4, 5. For problem 5, at the very least, you should answer all of the questions posed in
question 4 about the periodic Schroedinger equation...

The 12th annual **Canadian Undergraduate Mathematics Conference**
will be held at Queens University this year. July 13-17, 2005.
Check out their webpage and see
if you might like to participate!