REAL ANALYSIS
MAT 1000Y (MAT 457Y)
Course topics
- Lebesgue integration, measure theory, convergence theorems, the
Riesz representation theorem, Fubini's theorem, complex measures
- L^p-spaces, density of continuous functions, Hilbert space,
weak and strong topologies, integral operators.
- Inequalities.
- Bounded linear operators and functionals. Hahn-Banach theorem,
open-mapping theorem, closed graph theorem, uniform boundedness
principle.
- Schwartz space, introduction to distributions, Fourier transforms
on the circle and the line (Schwartz space and L^2).
- Spectral theorem for bounded normal operators.
Professor: Mary Pugh, mpugh@math. utoronto.ca
Office hours:
Mondays, 2:30-3:30, Tuesdays, 2:30-3:30 (until Tues April 8)
Office location: room 3141, Earth Sciences Centre,
22 Russell Street
Grader: Ching-Nam Hung, cnhung@math. utoronto.ca
Office location: room 4052, Sidney Smith,
(416) 978-3484.
Textbooks:
Kolmogorov, A.N. and Fomin, S.V., "Introductory Real Analysis", 1975.
There will be a second text, to be chosen later.
References:
Folland, G.B., "Real Analysis: Modern Techniques and their Applications"
Lieb, E.H. and Loss, M., "Analysis"
Royden, H.L., "Real Analysis"
Rudin, W., "Real and Complex Analysis"
Rudin, W., "Functional Analysis"
Taylor, A.E., "Introduction to Functional Analysis"
Torchinsky, A., "Real Variables"
Yoshida, K., "Functional Analysis"
Zimmer, R.J., "Essential Results of Functional Analysis"
I have put all but one of the above books on reserve at the Math-Stat
library in the basement of Sidney Smith. (The university doesn't currently have Zimmer's book. It's
a lovely book and costs only $29.54 at
amazon.ca) You cannot check the
books out --- you have to use them while in the library. The
library's hours are Monday to Friday, 9:00-5:00. Please let me know
if the restrictions on their use is a real problem for you.
Looking to buy used books? I've had good luck with
abebooks.com.
Your course mark will be based on homework (worth 20%), three term
exams (worth 15% each), and one final exam (worth 35%).
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Your first homework assignment, due Wednesday September 18.
The solutions to eight of the warm-up exercises were penned by
by Steven Sloot and Jacob Sone. (Thanks!!)
Here they are, please let me know if you find any problems with them.
Lecture Notes: Sept 13, 2002.
Lecture Notes: Sept 16, 2002.
Lecture Notes: Sept 18, 2002.
Lecture Notes: Sept 20, 2002.
Your second homework assignment, due Friday September 27.
Here are the
solutions.
The solutions to eleven of the warm-up exercises were penned by
by Sajiya Jalil and Carina Siu. (Thanks!!)
Here they are, please let me know if you find any problems with them.
Lecture Notes: Sept 23, 2002.
Lecture Notes: Sept 25, 2002.
Lecture Notes: Sept 27, 2002.
Your third homework assignment, due Friday October 4.
Here are the
solutions. Note: there's a mistake in the solution to problem 5a. Because the
matrices are _real_ valued, we cannot diagonalize when the eigenvalues
are complex. If the eigenvalues are a +/- i b, then the reduced form of
the matrix will be A_11 = a, A_12, = b, A_21 = -b, A_22 = a.
Lecture Notes: Sept 30, 2002.
Lecture Notes: Oct 2, 2002.
Lecture Notes: Oct 4, 2002.
See the Tietze Extension Theorem in Action! Just save the
file
tietze_extension.m in your home directory, start up matlab
(in your home directory) and type "tietze_extension" at the prompt. If
you're a hacker, open it up and change the function f and the number of
approximants and whatever...
Your fourth homework assignment, due Wednesday October 16.
Here are the
solutions.
Lecture Notes: Oct 7, 2002.
Remember -- no class on 10/9 and 10/11! I'll schedule make-up classes
later.
Google is our friend.
Here are some
topology notes
from Kazuo Yokoyama of Sophia University in Japan.
Here are some
topology notes
from Michael Van Opstall of the University of
Washington. (Please let me know if you find mistakes in either document.)
Lecture Notes: Oct 16, 2002.
Rudin's user-friendly proof of Stone-Weierstrauss when the domain is a closed
interval in the real line.
Royden's proof of Stone-Weierstrauss when the domain is a compact metric space or Banach space.
See the Stone Weierstrauss approximants in Action! Just save the
file
stone_weierstrauss.m in your home directory, start up matlab
(in your home directory) and type "stone_weierstrauss" at the prompt. If
you're a hacker, open it up and change the function f and the number of
approximants and whatever...
Lecture Notes: Oct 18, 2002.
Your fifth homework assignment, due Friday October 25, is from
Kolmogorov and Fomin: problem 6 on page 128 and problems 1-4 on page 137.
Here are the
solutions.
Lecture Notes: Oct 21, 2002.
Lecture Notes: Oct 23, 2002.
Lecture Notes: Oct 25, 2002.
Your first
term test
was in class on Monday October 28.
It is worth
15% of your course mark.
Here are the
solutions.
Lecture Notes: Oct 30, 2002.
Your sixth homework assignment, due Friday, November 15. The solutions to problems 3-5 are at the end of the solutions to the second term test.
Lecture Notes: Nov 4, 2002.
Lecture Notes: Nov 6, 2002.
Lecture Notes: Nov 8 and Nov 11, 2002.
Google is our friend. Here are some
notes on topological vector spaces from
Paul Garrett
of the University of Minnesota.
Also, see Rudin's "Functional Analysis" pages 1-13.
Lecture Notes: Nov 13, 2002.
There will be no class or office hour on Friday November 15. I'll
schedule a make-up class later. You can hand the homework in on Monday
November 18. On November 14, my office hour will be 1-2 rather
than the usual time of 3-4.
Lecture Notes: Nov 18, 2002.
Lecture Notes: Nov 20 and 22, 2002.
Here is an example of a space X which is isometric
to its second dual X^**, but which is not isometric with X^** when you
use the natural mapping from X to X^**. The example was discovered
by R.D.James and published in 1951.
Your seventh homework assignment, due Monday, December 2, is from
Kolmogorov and Fomin: page 171 #12, page 183 #8, page 194 #7 and #9,
page 205 #2 and #3.
Lecture Notes: Nov 25, 2002.
Lecture Notes: Nov 27, 2002.
Caveat! Kolmogorov and Fomin toy with our emotions --- they introduce
"the weak topology on L*" in section 20.3 and then in section 2.4 they
tell us... really, the topology in 20.3 is the weak* topology on L*, and now
we're going to define the weak topology on L*. So in these notes, my
references to the weak toplogy on L* are really the weak* topology. (All
of this was explained in class.)
There will be no class on Wednesday December 4. I'll schedule a
make-up class later. There will be no office hours on Monday December
2 or Thursday December 5.
Lecture Notes: Dec 2 and Dec 6, 2002.
For the Spring semester, class will meet twice a week instead
of three times a week. Starting Monday January 6, we will meet
on Mondays 12:40-2:00 in SS 5017A and on Wednesdays 12:10-1:30 in
UC 67.
We have 5 1/2 hours of class time to make up: I deferred class
four times in the fall (Oct 9, Oct 11, Nov 15, Dec 4) and will defer
it once in the spring (Jan 22). So we have five and a half hours of
classtime to make up. I will do this by teaching 12:10-2:00
on the following eleven Wednesdays: Jan 15, Jan 29, Feb 5,
Feb 12, Feb 26, Mar 5, Mar 12, Mar 19, Mar 26, Apr 2, Apr 9.
Note:
this is every Wednesday in the Spring semester except for Jan 8
(second term exam), Jan 22 (class deferred), and Feb 19 (reading
week). So the effective schedule for the Spring semester is M
12:40-2:00 in SS 5017A and W 12:10-2:00 in UC 67.
Lecture Notes: Jan 6, 2003.
Your second
term test
was in class on Wednesday January 8.
It is worth
15% of your course mark.
Here are the
solutions.
Lecture Notes: Jan 13, 2003.
Lecture Notes: Jan 15, 2003.
Your eighth homework assignment, due Wednesday, January 29.
Lecture Notes: Jan 20, 2003.
Remember! No class on Wednesday January 22!
Lecture Notes: Jan 27, 2003.
Lecture Notes: Jan 29, 2003.
Your ninth homework assignment, due Friday, February 14. Please put it
into Ching-Nam Hung's mailbox by 4 pm.
Here are the
solutions.
Lecture Notes: Feb 3, 2003.
Lecture Notes: Feb 5, 2003.
Lecture Notes: Feb 10, 2003.
Lecture Notes: Feb 12, 2003.
The book for the measure theory and integration portion of the course
will be Daniel W. Stroock's "A Concise Introduction to the Theory
of Integration". It is in stock in the UofT bookstore.
Lecture Notes: Feb 24, Feb 26, and March 3, 2003.
Here are some
practice problems on compact operators
and spectral theory for you to work in to help you prepare for the term exam.
The third
term test
was in class on Wednesday March 5. It
covered distributions, linear operators, and spectral theory.
Lecture Notes: March 10, 2003.
Lecture Notes: March 12, 2003.
Lecture Notes: March 19, 2003.
Lecture Notes: March 24, 2003.
Lecture Notes: March 26, 2003.
Lecture Notes: March 31, 2003.
Lecture Notes: April 2, 2003.
Lecture Notes: April 7 and 9, 2003.
Extra office hours before final exam
The final exam will be on Friday May 2 from 10 am to 1 pm in SS 1083.
You're allowed a "cheat sheet" for the exam: 8.5" x 11", with writing
on one side. You can put definitions and statements of theorems on
it. No proofs or sketches of proofs or counter-examples. You'll hand
the cheat sheet in with your exam. If you have specific questions
about what's fair game for the cheat sheet, please email me.
Ching-Nam Hung will have twenty hours of office hours the week before the
exam. His office is SS 4052. His office hours will be:
Friday Apr 25 1-5pm,
Monday Apr 28 1-5pm,
Tuesday Apr 29 2:30-6:30pm,
Wednesday Apr 30 9am-1pm, and
Thursday May 1 9am-1pm.
I will have office hours on Monday April 28 1pm-4pm.
Here are some old UofT
comprehensive exams.
Here are some old UIUC
comprehensive exams.
Here are some review notes, courtesy of
David Rose of UIUC:
set 1,
set 2,
set 3, and
set 4.