MAT 550: Analysis (Spring 2009)
Department of Mathematics
SUNY at Stony Brook
Text: A First Graduate Course in Real Analysis,
by Daryl Geller, available only from Donna, the Graduate secretary.
Several lectures will be on additional topics in PDE; supplementary
notes will be distributed.
Professor: Prof. Daryl Geller, Math Tower 4-100B
Phone: 632-8327 email: daryl@math.sunysb.edu
Office hours: Tuesdays and Thursdays, 2:30-4, in 4-100B.
Homework: Homework will be assigned each week, and will
count for 15% of your grade in the course.
Examinations: There will be two tests during the semester
(dates to be negotiated). Together, they will count for 50%
of
the grade in the course. The final examination will count for 35%
of the grade in the course.
Final Exam -- Thursday, May 14, 4-130 Math (our regular classroom), 11-1:30 p.m.
Homework and announcements will be posted on this page regularly.
Announcements for the week of 1/26-1/30:
Homework (due Thursday Feb 5) is:
2008 edition: Exercises 7.1.6, 7.1.7, 7.3.4, 8.3.4, 8.3.5.
2004 edition: Exercises 7.1.1, 7.1.2, 7.3.1, 8.3.1, 8.3.2.
Also read the definition of Lp(X) on the first page of
Chapter 9, and do this problem from the January 2009 comps:
(a) Say p is finite and p > 1. Exhibit, with proof, a function f which
is in L1(0,1), which is not in Lp(0,1).
(b) Exhibit, with proof,
a function F which
is in L1(0,1), which is not in Lp(0,1) for any p >
1. (Hint: try to define F piecewise, using the functions of part (a).)
Announcements for the week of 2/2-2/6:
Homework (due Thursday 2/12) is:
2008 edition: Exercises 9.1.7, 9.1.8, 9.1.12, 9.1.13, 9.1.14, 9.2.8,
9.2.9, 9.2.10.
2004 edition: Exercises 9.1.1, 9.1.2, 9.1.3, 9.1.4, 9.1.5, 9.2.1, 9.2.2,
9.2.3.
More problems than usual, but they are all pretty straightforward, and
instructive.
Announcements for the week of 2/5-2/9:
Homework (due Thursday 2/19) is:
2008 edition: Exercises 9.3.8, 9.5.9, 9.5.10,
9.5.11, 9.5.12, 9.5.13, 9.6.2.
2004 edition: Exercises 9.3.3, 9.5.3, 9.5.4, 9.5.5, 9.5.6, 9.5.7, 9.6.1.
Also show that L&infin(0,1) is not equal to &cap 1 < p
< &infin
Lp(0,1).
Announcements for the week of 2/16-2/20:
Homework (due Thursday 2/26) is:
2008 edition: Exercises 10.2.9, 10.2.10, 10.2.11, 10.3.13, 10.3.16.
2004 edition: Exercises 10.2.4, 10.2.5, 10.2.6, 10.3.3, 10.3.5.
Also show that if 1 ≤ p ≤ &infin, then the closed unit ball of
Lp(0,2&pi) is not compact. (Hint: show that if
en(x) = einx, then the en have
no convergent subsequence. The easiest case is p=2.)
Announcements for the week of 2/23-2/27:
First midterm:Tuesday 3/3, on Chapters 7-10, in class.
Homework (due Friday 3/6, by noon, under my office door) is:
2008 edition: Exercises 11.1.1, 11.1.2, 11.2.5, 11.2.6
2004 edition: Exercises 11.1.1, 11.1.2, 11.2.2, 11.2.3
In problem 11.1.1, explain the differences between the seasons as
quantitatively as possible (in other words, give formulas; you don't
have to numerically evaluate anything).
Also, answer this question, using the ideas of section 11.1:
It is raining, and a person heads for a shelter straight ahead. Does he
get less wet if he walks or if he runs? Assume the rain is falling
straight down, and that the person has the shape of a rectangular block,
so that he gets wet only on his top and on his front.
Announcements for the week of 3/2-3/6:
Solutions for the first midterm are available
here.
The highest grade was 100. Grades under 40 are cause for concern.
Homework (due Thursday, 3/12, in class) is:
2008 edition: Exercises 10.3.12, 10.3.15, 11.2.13, 11.2.27, 11.3.1
2004 edition: Exercises 10.3.2, 10.3.4, 11.2.4, 11.2.11, 11.3.1.
Announcements for the week of 3/9-3/13:
After we finish chapter 11, we will cover some supplementary notes
on the Fourier transform and the heat equation. Those supplementary
notes are
here.
Homework (due Thursday, 3/19, in class) is:
2008 edition: Exercises 11.2.19, 11.2.20, 11.2.21, 11.2.24. 11.2.25,
11.3.2, 11.3.6
2004 edition: Exercises 11.2.5, 11.2.6, 11.2.7, 11.2.8, 11.2.9, 11.3.2,
11.3.6.
Announcements for the week of 3/16-3/20:
As I said in class, I would like to reschedule the Thursday, April 30 class
for Friday, April 3, right before the break. One student asked that we
have class before Algebra II -- that is, from 9:10-10:30. That is fine
with me. If this would cause problems for anyone, please let me know soon.
I fixed the supplementary notes. Those notes are
here.
Homework (due Thursday, 3/26, in class) is:
2008 edition: Exercises 11.4.4, 11.4.5, 11.4.8, 11.4.9.
2004 edition: Exercises 11.4.2, 11.4.3, 11.4.4, 11.4.5.
Also do Exercises 1.2 and 1.6 of the supplementary notes.
Also do
this exercise.
Announcements for the week of 3/23-3/27:
Next week we have an extra class at 9:10 a.m. on Friday, in the usual
room.
Next week we will cover sections 12.3-12.5. We will skip sections
12.1-12.2 for now, and come back to them later.
The second midterm will be on Tuesday, April 21, in class. It will
cover Chapter 11, the supplementary notes, and sections 12.3-12.5.
Homework (due Thursday, 4/2, in class) is:
Supplementary notes: Exercises 6.4 and 6.6.
Also do
these exercises
(fixed Friday night after someone noticed a typo).
The proofs of Corollary 2.3 and Theorem 4.3 in the supplementary notes
needed repair. The repaired notes are
here.
Announcements for the week of 3/30-4/3:
On Friday April 3, we finished section 12.5 and also did most of section
12.1.
Homework (due Thursday, 4/16, in class) is:
(2008 edition): Exercises 12.3.11, 12.3.16, 12.3.17, 12.3.18, 12.3.22, 12.3.23,
12.4.4, 12.4.5, 12.4.6, 12.4.8, 12.4.9, 12.5.9, 12.5.11, 12.5.12
(2004 edition): Exercises 12.3.3, 12.3.4, 12.3.5, 12.3.6, 12.3.10,
12.3.11, 12.4.2, 12.4.3, 12.4.4, 12.4.6, 12.4.7, 12.5.2, 12.5.3,
12.5.4
In problems where the fundamental maximal principle is to be used,
identify the maximal operator and try to use it, if appropriate.
Announcements for the week of 4/13-4/17:
Today in class (4/16) we got up to the end of section 13.2.
The second midterm will be on Tuesday, April 21, in class. It will
cover Chapter 11, the supplementary notes, and sections 12.3-12.5.
Half of the test will be about actually solving PDE, so please devote
sufficient focus to those sections of the book, and the supplementary
notes, where PDE are actually solved.
Homework (due Friday, 4/24, by 5 p.m., under my office door):
(2008 edition) Exercises 12.1.18, 12.1.19, 12.2.3, 12.2.13,
12.2.14.13.1.3, 13.1.6.
(2004 edition) Exercise 12.1.3, 12.1.4, 12.2.2, 12.2.6, 12.2.7, 13.1.1,
13.1.2.
Typos in the supplementary notes: 4 lines after equation (15), replace
"n" in the exponent by "1". Also in Exercise 6.6,
P should be
defined as U × {T}.
Before next Thursday's class, please review the material about
absolutely continuous functions, on pages 282-284.
Announcements for the week of 4/20-4/24:
Next week we WILL have our usual class on Tuesday, April 28. There will be no
class on Thursday, April 30, since I will be at a conference.
Your graded midterms are in your mailboxes. Solutions are
here..
The highest grade was 69. Grades under 35 are cause for concern.
The only remaining classes for this course are on Thursday May 7, first
from 8:20-10 a.m., and then from 11:20-12:40, both in our usual
room.
Homework (due Thursday, May 7, in class and no later:):
(2008 edition): Exercises 13.2.5, 13.2.6, 13.3.9, 13.3.10, 13.3.11,
13.4.3, 13.4.4, 13.5.10, 13.5.11, 13.5.12, 13.5.17, 13.5.22,
13.5.23.
(2004 edition): Exercises 13.2.1, 13.2.2, 13.3.2, 13.3.3, 13.3.4,
13.4.1, 13.4.2, 13.5.1, 13.5.2, 13.5.3, 13.5.4, 13.5.5, 13.5.6.
Also do problem 3 from the midterm, in the manner suggested there. (Do
not use conformal mapping. This does work, but the method suggested on
the midterm generalizes to the situation where x is in
Rn, not just R (still
y is greater than or equal to zero). Conformal mapping won't work in
that case.) Adapt the methods on pages 15-16 of the supplementary
notes.
Re the proof of Propositon 13.5.8 (a): in the last few lines of the
proof, the forms should all be restricted to BR+1; what they are
outside that ball is irrelevant.
Announcements for the week of 5/4-5/8:
Final Exam -- Thursday, May 14, 4-130 Math (our regular classroom),
11-1:30 p.m.
The exam will focus on Chapter 11, the supplementary notes, and sections
12.1, 12.2, 12.4, 13.1, 13.2, 13.3, and 13.4. Please also be sure that
you know the statements of all the main results in chapters 7 through
12. Again be prepared to actually solve PDE.
Because of the exam schedule for the other graduate courses, it looks
like it would be most useful if I were around on Wednesday, May
13. I'll be in my office from 2:15 until 5:40 on that day.
Please don't hesitate to visit me if you have questions.
The solutions for the second midterm, including problem 3, are
here..
DSS advisory. If you have
a physical, psychiatric, medical, or learning disability that may affect
your ability to carry out the assigned course work, please contact the
office of Disabled Student Services (DSS), Humanities Building, room 133,
telephone 632-6748/TDD. DSS will review your concerns and determine
what accommodations may be necessary and appropriate. All information and
documentation of disability is confidential.