# MAT 544: Analysis

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### <img SRC="logo4cxx.gif" ALT="SUNY at Stony Brook"  align=CE NTER>Department of MathematicsSUNY at Stony Brook <hr>

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Topics covered

New graduate students who feel that they do not need to take t his
course must get a waiver from the course professor.
Waivers will not
be unreasonably denied.

Text: A First Graduate Course in Real Analysis, Part One, by Daryl Geller,
available only at the University Bookstore.

Professor: Prof. Daryl Geller, Math Tower
4-1 00B

Phone: 632-8327 email: daryl@math.sunysb.edu

Office hours: TuTh 5:15-6:45 p.m.

Homework:  Homework will be assigned each week, and w ill count for
15% of your grade in the course.

Examinations: There will be two tests during the semester< BR>(dates to be negotiated).   Together, they will count for 50% o f the
grade in the course.  The final examination will count for 35% of the
grade in the course.

Final Exam -- Thursday, December 18, 11 a.m. - 1:30 p.m.

Errata in the book:

Page 20: replace the sen tence starting on the sixth last line: "We may let
i1 = N." by "Let i0 = N."

Page 70: replace Y0 o n this page, in the two places where it occurs, by YO'.

Page 96: six lines from the bottom: change "some y" to "some nonzero y" .

Page 102: in the proof of Lemma 3.1.2, it should have been n oted that (3.1) implies
that S(0) = 0.

Page 108: the defi nition of bj, near the bottom of the page, is wrong; multiply
what is written by ||T||j.

Page 111: in the second line from the bottom, ej has not been defined.  It
is the column vector which has a 1 in the jth slot and zeros elsewhere.

Page 133: in the first two paragraphs of the proof (before it says "Now we can reduce
to..."), change "T" to "S" everywhere, and "S" to "T" every where.

Page 147: In equation (3.69), r equals 1/2.

P age 197: In (4.17), add the words "for every epsilon > 0" at the begi nning.

Page 212: Third line from the bottom: replace "Proposit ion 4.3.1" by "Exercise 4.3.1".

Page 268: 5 lines from the bottom: this is the Lp norm of FN, not that norm raised to the pth power.

Page 260:Exercise 5.7.10: Assume that phi is continuous on the interior of I, but don't assume I is closed. In the hint, prove that the indicated point is in the CLOSURE of U(phi) (still a convex set). This is good enough, since the integral of f is in the interior of I unless f is constant a.e. (If f is constant a.e,, Jensen's inequality is trivial.) Delete exercise 5.7.11.

Please let me know if you spot any mo re mistakes.

Announcements for the week of 9/8-9/12:
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Ho mework (due Tuesday 9/16) is: Exercises 1.2.4, 1.2.5, 1.2.6, 1.3.2,
1.3.3 , 1.3.10.

Announcements for the week of 9/15-9/19:

Ho mework (due Tuesday 9/23): Exercises 1.4.4, 1.4.5, 1.4.6, 1.5.3,
2.1.2, 2 .1.3, 2.2.2 (see the definition on page 69).

Announcements for the week of 9/22-9/26:

Ho mework (due Tuesday 9/30): Exercises 2.2.4, 2.2.7, 2.2.8,
2.2.10, 2.3.1, 2.3.2.  Please note that 2.2.7
is difficult; if you think it's easy, you've made a mistake.

Announcements for the week of 9/29-10/3:

Ho mework (due Tuesday 10/7): Exercises 2.3.4, 2.3.5, 2.3.6 (these
three are essentially one exercise), 2.3.7 (rather tricky) 2.3.8,
3.1.2 and 3.1.5.

Announcements for the week of 10/6-10/10:

Ho mework (due Tuesday 10/14): Exercises 3.2.5, 3.3.4, 3.3.5, 3.3.6.

First midterm: In class, on Thursday, October 9.  It will cover
up to the end of section 3.2.

Announcements for the week of 10/13-10/17:

Ho mework (due Tuesday 10/21): Exercises 3.6.5, 3.6.7, 3.6.8, 3.6.9.

Announcements for the week of 10/20-10/24:

Ho mework (due Tuesday 10/28): Exercises 3.8.2, 3.8.3, 3.8.5, 3.8.6
(these w ill get you used to the ideas), 3.8.7, 3.8.8, 3.8.9 (these are
more subst antial).

Announcements for the week of 10/27-10/31:

Ho mework (due Tuesday 11/4): Exercises 4.1.2, 4.2.2.  Also please
read Definitions 4.3.1, 4.3.4 and 4.3.5, and do Exercises 4.3.3, 4.3.4,
4.3.5 and 4.3.6.

Announcements for the week of 11/3-11/7:

Ho mework (due Tuesday 11/11): Exercises 4.5.1, 4.5.2, 4.5.3. 4.5.4,
and 4.5 .5.

Announcements for the week of 11/10-11/14:

Ho mework (due Tuesday 11/18): Exercises 4.6.2 (just use basic properties
of measure, not the (A)(B)(C) method), 4.6.5 (a)(b)(c), 4.7.1, 4.7.2, 4.7.3,4.7.4.

Announcements for the week of 11/17-11/21:

Ho mework (due Tuesday 11/25): Exercises 5.2.1 (this is actually 3
related exercises),
5.2.4, 5.2.6, and 5.3.1.  Also, begin reading section 6. 2 and do problems
6.2.1 and 6.2.2.

Second midterm: Thursday November 20, 7-10 p.m.  Ther e will
be no class on that day (I'll be away, so there won't be any offic e hours
either).  Midterm covers from the beginning of section 3.3 t o the end of
chapter 4.  Location: Physics P-124.
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Announcements for t he week of 11/24-11/28.
Sorry, I'm still away. Thanks for your i ndulgence. I hate missing classes, but this time it's unavoidable. We will s till be able to finish the syllabus.
Homework (Due 12/2/03): Exercises 6. 2.3, 6.2.4, 6.2.5, 6.2.6, 6.2.7, 6.2.8. Also please read: (1) the statement of theorem 5.3.4 (MCT) (you don't have to read the proof) (2) the statement of Proposition 5.3.6 and its easy proof. Please do exercises 5.3.2 and 5.3.3 . Happy Thanksgiving!

Announcements for t he week of 12/1-12/5.
Sorry this web page got so messed up. It's legible now. Homework (due Tuesday 12/9): Exercises 5.3.4, 5.3.5, 5.3.6, 5.5.2, 5.6.2, 5.6.3.

FINAL HOMEWORK (due Friday 12/19, but you should try to do it earlier!) Exercises 5.7.4, 5.7.5, 5.7.7, 5.7.8, 5.7.9, 5.7.10.

FINAL EXAM -- Thursday, December 18, 11:00-1:30 p.m. in our regular classroom (Physics P-129). Covers chapters 4 and 5, plus sections 6.1, 6.2, 1.3, 2.3, 3.4, 3.5 and 3.6. I'll be in my office and able to answer questions on Tuesday, December 16, from 2-5 p.m.

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 chapters 3 and 6 and the "Postponed Proofs" sections in Chapters 7 and 8 are optional reading (you are not going to be tested on them, unless Professor Shafikov wants you to learn parts of these sections), 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, psych iatric, medical, or learning disability that may affect