MAT 544: Analysis

SUNY at Stony Brook
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 at the University Bookstore.

Professor: Prof. Daryl Geller, Math Tower 4-100B
Phone: 632-8327 email:
Office hours: to be announced, in 4-100B or P-143.

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 -- Tuesday, December 19, 11 a.m. - 1:30 p.m.

Homework and announcements will be posted on this page regularly.
Announcements for the week of 9/11-9/15:
Homework (due Thursday 9/21 ) is: Exercises 1.2.7, 1.2.13, 1.3.9, 1.3.11. Also, for Tuesday 9/26, please do Exercise 1.2.10 (sorry for the earlier typo).

Announcements for the week of 9/18-9/22:
Homework (due Thursday 9/28) is: Exercise 1.2.10 (see above), 1.3.30, (1.4.6, 1.4.7, 1.4.8) (this is basically one problem), 1.4.9, 1.5.8, 1.5.9.

Announcements for the week of 9/25-9/29:
Homework (due Thursday 10/5) is: Exercises 2.1.1, 2.1.8, 2.1.10, 2.2.12, 2.2.18 (not easy!), 2.2.19

Announcements for the week of 10/2-10/6:
Homework (due Thursday 10/12) is: Exercises 2.2.20; 2.2.21; [2.3.22, 2.3.23, 2.3.24] (this is basically one exercise); 2.3.25 (try to think geometrically); 2.3.26.

Announcements for the week of 10/9-10/13:
Homework (due Thursday 10/19) is: 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.

First midterm: Tuesday October 24, 7:30-10:30 p.m., in P-131 Math Building. It will cover up to the end of page 134 in the book.
Announcements for the week of 10/16-10/20:
Homework (due Tuesday 10/31) is: Exercises 3.6.3. 3.6.5, 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.

Announcements for the weeks of 10/23-11/3:
Homework (due Thursday 11/9) is: Exercises 3.8.3, 3.8.4, (these are just to help you understand the proof of the Implicit Function Theorem), 3.8.7, 3.8.8, 3.8.9, 3.8.10, 4.1.14, 4.1.18, 4.1.22, 4.2.4.
Solutions to the first midterm are here. The first part of the first set of supplementary notes is here.

Announcements for the weeks of 11/6-11/10:
Homework (due Tuesday 11/14) is: Exercises 0.3 and 0.4 of the first set of supplementary notes.
Homework (due Thursday 11/16) is: Exercises 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), 4.3.14, 4.3.15, 4.3.16, 4.3.17, and the following additional exercise:
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).)

Announcements for the weeks of 11/13-11/17:
The first set of supplementary notes is finished. It is here.
The second set of supplementary notes is here.
Homework for Tuesday, November 20: Exercise 0.1 of the second set of supplementary notes (there was a typo in the statement of this problem, which has now been fixed); Exercises 4.5.6, 4.5.7, 4.5.8, 4.5.9, 4.5.10.
We tentatively scheduled the second midterm for the evening of Thursday, November 30. If anyone has a problem with that, please send me an E-mail right away.

Announcements for the week of 11/20-11/22:
Homework (due Thursday 11/30) is: Exercises 4.6.5, 4.6.13(a)(b)(c), 4.7.3, 4.7.4, 4.7.5, 4.7.6
Second midterm: Thursday, November 30, 7-10 p.m., in 5-127 Math. The test will be about the material on pages 134-218.

Announcements for the week of 11/27-12/1:
Homework (due Thursday 11/7) is: Exercises 5.1.6, 5.2.6, 5.2.14, 5.2.21, 5.3.8, 5.3.9.
Solutions to the second midterm are here.
The remaining homework assignments are:
Homework due Thursday 12/14: Exercises 5.3.14, 5.5.8, 5.6.5, 5.6.7, 5.7.10, 5.7.11, 5.7.12, 5.7.13, 5.7.14.
Homework due Thursday 12/21: Read section 6.2, and do Exercises 6.2.1, 6.2.2, 6.2.3, 6.2.4, 6.2.5, 6.2.6, 6.2.7, 6.2.8, and 6.2.9. (You can start doing 6.2.1 through 6.2.8 now if you like since we have already covered the prerequisite material. Hint for exercise 6.2.2 (b): the answer is no.)
The third set of supplementary notes (on Lebesgue-Stieltjes integrals) is here.

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.

FINAL EXAM -- Tuesday, December 19: 11:00 a.m. - 1:30 p.m., in our regular classroom (Harriman 115). It will cover chapters 4, 5 and 6, section 2.3, sections 3.6, 3.7, 3.8, and the second set of supplementary notes. Of course Chapter 1 is fundamental.

Errata in the book:
page 70, line 12: Replace the sentence
Select K, epsilon > 0 as in (2.12).
Select K, epsilon > 0 as in the analogue of equation (2.12) where one uses Y0' in place of Y0.
page 92, bottom picture: translate by -u, not -v.
page 109, line 13, third character: V not v

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.