MAT 544: Analysis

<center>

<img SRC="logo4cxx.gif" ALT="SUNY at Stony Brook" align=CE NTER>

Department of Mathematics

SUNY at Stony Brook

<hr>

**New graduate students who feel that they do not need to take t
hiscourse 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

i_{1} = N."
by "Let i_{0} = N."**Page 70:** replace Y_{0} o
n this page, in the two places where it occurs, by Y_{O}'.**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 b_{j}, near the bottom of the page, is wrong; multiply

what is written by ||T||^{j}.**Page 111:** in the second
line from the bottom, e_{j} 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 L^{p} norm of F_{N}, 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:**

<
/FONT>

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 problems6.2.1 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.

<
/FONT>
**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!

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

your ability to c
arry out the assigned course work, please contact the

office of Disabled
Student Services (DSS), Humanities Building, room 133,

telephone: 632-674
8/TDD. DSS will review your concerns and determine

what accommodations ma
y be necessary and appropriate. All information and

documentation of disa
bility is confidential.**
**