MAT 544: Analysis (Fall 2008)
Department of Mathematics
SUNY at Stony Brook
New graduate students who feel that they do not need to take this
course must get a waiver from the course professor. Waivers
will not be unreasonably denied.
Text: A First Graduate Course in Real Analysis,
by Daryl Geller, available only in the Graduate Math Office.
Professor: Prof. Daryl Geller, Math Tower 4-100B
Phone: 632-8327 email: daryl@math.sunysb.edu
Office hours: For the first part of the semester: Fridays, 11-2.
Homework: Homework will be assigned each week, and will
count for 15% of your grade in the course. Please hand it in, in class,
on Wednesdays.
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 -- Wednesday, December 17, 2 - 4:30 p.m.
Homework and announcements will be posted on this page regularly.
Announcements for the week of 9/2-9/5:
Homework (due Monday 9/8) is: Exercises 1.1.3, 1.1.4, 1.1.12, 1.1.15,
1.1.23, 1.1.24.
Announcements for the week of 9/8-9/12:
Homework (due Wednesday 9/17) is:
(2008 edition): Exercises 1.1.28, 1.2.7, 1.2.10, 1.2.13, 1.3.9, 1.3.11
(2004 edition): Exercises 1.1.11, 1.2.1, 1.2.4, 1.2.5, 1.3.2, 1.3.3
Announcements for the week of 9/15-9/19:
Homework (due Wednesday 9/24) is:
(2008 edition): Exercises 1.3.30, (1.4.6, 1.4.7, 1.4.8) -- [this is
basically one problem], 1.5.8, 2.1.1, 2.1.8, 2.1.10
(2004 edition): Exercises 1.3.10, (1.4.4, 1.4.5, 1.4.6) -- [this is
basically one problem], 1.5.3, 2.1.1, 2.1.4, 2.1.6
Announcements for the week of 9/22-9/26:
Homework (due under my door 4-100B by Friday 10/3 at 8 a.m.) is:
(2008 edition): Exercises 2.2.12, 2.2.18 (not easy!), 2.2.19, 2.2.20,
2.2.21, 2.3.11
(2004 edition): Exercises 2.2.3, 2.2.7 (not easy!), 2.2.8, 2.2.9,
2.2.10, 2.3.2
Announcements for the week of 9/29-10/3:
Homework (due Wednesday 10/8) is:
(2008 edition): Exercises 1.4.9, 1.5.9, 2.3.14, (2.3.22, 2.3.23.
2.3.24)--(this is basically one exercise), 2.3.25 (try to think
geometrically), 2.3.26
(2004 edition): Exercises 1.4.7, 1.5.4, 2.3.3, (2.3.4, 2.3.5.
2.3.6)--(this is basically one exercise), 2.3.7 (try to think
geometrically), 2.3.8
Announcements for the week of 10/6-10/10
Homework (due Wednesday 10/15) is:
(2008 edition): Exercises 3.1.9, 3.1.13, 3.2.1, 3.2.9, 3.3.6, 3.3.7,
3.3.8, 3.5.8
(2004 edition): Exercises 3.1.2, 3.1.5, 3.2.1, 3.2.5, 3.3.4, 3.3.5,
3.3.6, 3.5.2
Announcements for the week of 10/13-10/17
Homework (due Wednesday 10/22) is:
(2008 edition): Exercises 3.6.3, 3.6.12 (work
in the Banach space setting: that is, prove Proposition 3.9.1 on page
163 as well); 3.6.16, 3.6.20, 3.6.21.
(2004 edition): Exercises 3.6.1, 3.6.5
(work in the Banach space setting: that is, prove Proposition 3.9.1 on page
163 as well); 3.6.7, 3.6.8, 3.6.9.
First midterm: Monday, October 27, in class, covering up to the end of
Chapter 3.
Homework due Monday October 27 is:
(2008 edition) Exercises 3.8.3, 3.8.4, 3.8.6
(2004 edition) Exercises 3.8.2, 3.8.3, 3.8.5
These exercises will help you study for the test.
Announcements for the week of 10/20-10/24
Homework (due under my door 4-100B by Friday 10/31 at 12 p.m.) is:
(2008 edition): Exercises 3.8.7, 3.8.8, 3.8.9, 3.8.10, 4.1.14, 4.2.6
(in (a), prove more
generally that if V is any open set contained in [0,1], with m(V)=1,
then the characteristic function of V is Riemann integrable).
(2004 edition): Exercises 3.8.6, 3.8.7, 3.8.8, 3.8.9, 4.1.1, 4.2.2
(in (a), prove more
generally that if V is any open set contained in [0,1], with m(V)=1,
then the characteristic function of V is Riemann integrable).
First midterm: Monday, October 27, in class, covering up to the end of
Chapter 3.
Announcements for the week of 10/27-10/31
I will be out of town on Wednesday, November 5. I will hold a
makeup class on Friday, November 7, from 10:40 a.m.-12 p.m., in our
usual classroom.
Homework (due Friday, November 7, in class) is:
Define a function f, mapping the reals to the reals, by the rule
f(x) = 3x for x less than 1/2, f(x) = 3(1-x) for x greater than or equal to 1/2.
Find the (filled-in) Julia set of f (that is, the set of real numbers whose
iterates under f don't approach plus or minus infinity). (Hint: use
Exercise 4.2.6 (b) of the 2008 edition, which is Exercise 4.2.2 (b) of
the 2004 edition.) Also:
(2008 edition): Exercises 4.3.14, 4.3.15, 4.3.16, 4.3.17 (these will be used
on Monday, so please try to do them by then!) Also 4.1.23 and 4.3.4.
(2004 edition): Exercises 4.3.3, 4.3.4, 4.3.5, 4.3.6 (these will be used
on Monday, so please try to do them by then!) Also 4.1.4 and 4.3.1.
Announcements for the week of 11/3-11/7
Best election story I heard: A man was waiting on line to vote, and
someone asked him how long he had been waiting. "About 200 years,"
he replied.
Our second midterm will be on class on Monday, November 17. It will
cover Chapter 4.
Homework (due Friday, November 14, by 12 noon, under my office door) is:
(2008 edition): Exercises 4.5.6, 4.5.7, 4.5.8, 4.5.9, 4.5.10,
4.6.6, 4.6.13 (a)(b)(c).
(2004 edition): Exercises 4.5.1, 4.5.2, 4.5.3, 4.5.4, 4.5.5, 4.6.2,
4.6.5(a)(b)(c).
Announcements for the week of 11/10-11/14
Our second midterm will be on class on Monday, November 17. It will
cover Chapter 4.
Homework (due Friday, November 21, by 12 noon, under my office door) is:
(2008 edition): Exercises 4.7.3, 4.7.4, 4.7.5, 4.7.6, 5.2.6, 5.2.14
(2004 edition): Exercises 4.7.1, 4.7.2, 4.7.3, 4.7.4, 5.2.1, 5.2.4
Announcements for the week of 11/17-11/21
The midterms are in your mailboxes. Official solutions should be in your
mailboxes on Friday. The highest grade was 97. Grades under 35 are cause for
concern.
Homework (due Wednesday, November 26, in class) is:
(2008 edition): Exercises 5.2.21, 5.3.8, 5.3.9, 5.3.13, 5.3.14; also read
pages 276-277 and do Exercises 6.2.3 and 6.2.4. (Hint for exercise 6.2.4
(b): the answer is NO.)
(2004 edition): Exercises 5.2.6, 5.3.2, 5.3.3, 5.3.5, 5.3.6; also read
pages 276-277 and do Exercises 6.2.1 and 6.2.2. (Hint for exercise
6.2.2 (b): the answer is NO.)
Announcements for the week of 11/24-11/28
The remaining homework assignments are:
Due Wednesday, December 3, in class:
(2008 edition): 5.5.8, 5.6.5, 5.6.7, 5.7.10, 5.7.11, 5.7.12, 5.7.13.
(2004 edition): 5.5.2, 5.6.2, 5.6.3, 5.7.3, 5.7.4, 5.7.5, 5.7.6.
Due Wednesday, December 10, in class:
(2008 edition): 5.7.14, 5.7.15, 5.7.16, 5.7.17. Also read pages
278-279, and do 6.2.6 and 6.2.7.
(2004 edition): 5.7.7, 5.7.8, 5.7.9, 5.7.10. Also read pages 278-279,
and do 6.2.3 and 6.2.4.
Due Wednesday, December 17, at the exam: (but you can do them already,
since we have already covered the prerequisite material):
(2008 edition): Read pages 280-284, and do 6.2.8, 6.2.9, 6.2.12, 6.2.13
and 6.2.17.
(2004 edition): Read pages 280-284, and do 6.2.5, 6.2.6, 6.2.7, 6.2.8
and 6.2.9.
Intersession homework: read chapters 7 and 8. These chapters
will be covered rapidly at the beginning of MAT 550.
Section 6.3, the "Suggested
Intersession Reading" in chapter 6 and the "Postponed Proofs" sections
in Chapters 7 and 8 are optional reading (you are not going to be tested
on them, but they are HIGHLY RECOMMENDED).
Before leaving for intersession, please
go to the math library, and make a copy of Doss's one-page
proof of the Hahn Decomposition Theorem, in the
Proceedings of the American Mathematical Society, of October
1980; you will want to have this when reading chapter 8.
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.