MAT 544: Analysis

Department of Mathematics

SUNY at Stony Brook

Phone: 632-8327 email:

Final Exam --

Homework and announcements will be posted on this page regularly.

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).

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.

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

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.

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.

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.

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.

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).)

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.

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

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

Homework due

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.

page 70, line 12: Replace the sentence

Select K, epsilon > 0 as in (2.12).

by

Select K, epsilon > 0 as in the analogue of equation (2.12) where one uses Y

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.