Math 4331, Section 08610 - Fall 2000

Introduction to Real Analysis

Classes: 4:00-5:30pm MW, in 348 PGH
Instructor: Andrew Török
Office: 672 PGH
Phone: (713) 743-3478
E-mail: torok@math.uh.edu
Office hours: see home-page

This is the first half of a two-semester course. The prerequisite is Math 3333 or 3334.

Note:

a good book for Math 3333 is "Analysis with an introduction to proof" by Steven R. Lay.

another book used for Math 3333 is "Elementary analysis : the theory of calculus" by Kenneth A. Ross.

one of the books used for Math 3334 is "Calculus on manifolds; a modern approach to classical theorems of advanced calculus" by Spivak, Michael.

another recommended book for Math 3334 is "Vector calculus" by Jerrold E. Marsden and Anthony J. Tromba.

The book we will be using this year is

Principles of Mathematical Analysis

The intention is to cover in two semesters most of the book.

Some of the topics are: topology of metric spaces, limits and continuous functions, infinite series, partial differentiation, the basic theorems of analysis (local inversion, implicit functions), integrals, integration on "surfaces" and Stokes' theorem.

The course will place a lot of emphasis on proofs. Most of the homework will be assigned from the textbook.

You can find here a very good set of web-notes for analysis on the real line (courtesy of Prof. John Orr, Univ. of Nebraska-Lincoln). Warning: it involves a lot of Java activity.

As we advance, I am going to post the reading assignments and the homeworks.
A ^* denotes more difficult problems.

HOMEWORKS
Chapter HAND-IN (due date) Solve only, do not hand-in

1 2, 5 (Aug. 30) 1, 4, 9, 12, 13, 14, 15, 17, 19

2 2, 3, 4 (Sept. 6; assume R is uncountable)
6 (Sept. 13)
two problems handed out in class (Sept. 27)
22, 29 (Oct. 4; you have to prove that Q is dense in R) 5, 8, 10,
9, 12, 14, 20,
7, 13
15, 16, 26 (including 23, 24)
21
25

3 1, 2 (Oct. 11)
3, 20 (Oct. 18)
5, 7 (Nov. 1)
13, 14 a, b, c (Nov. 8; extended to Nov. 13) 16 - 18, 21, 22, 24^* - 25

6-14
all problems except #19
4 2, 3 (Nov. 15) 1, 4, 5, 7, 20, 22, 23, 24
5 1, 12, 7, 13,
4 (see Thm. 5.8),
2^*, 3, 5, 6, 8^*, 9, etc. (use Thm. 5.10)

WARNING: please do not use arguments like "clearly", "obviously" etc. in your solutions! You might lose points if you do not motivate your steps and conclusions.

HOMEWORKS
1	2, 5 (Aug. 30)	1, 4, 9, 12, 13, 14, 15, 17, 19
2	2, 3, 4 (Sept. 6; assume R is uncountable) 6 (Sept. 13) two problems handed out in class (Sept. 27) 22, 29 (Oct. 4; you have to prove that Q is dense in R)	5, 8, 10, 9, 12, 14, 20, 7, 13 15, 16, 26 (including 23, 24) 21 25
3	1, 2 (Oct. 11) 3, 20 (Oct. 18) 5, 7 (Nov. 1) 13, 14 a, b, c (Nov. 8; extended to Nov. 13)	16 - 18, 21, 22, 24^* - 25 6-14 all problems except #19
4	2, 3 (Nov. 15)	1, 4, 5, 7, 20, 22, 23, 24
5		1, 12, 7, 13, 4 (see Thm. 5.8), 2^, 3, 5, 6, 8^, 9, etc. (use Thm. 5.10)

On Tuesday, Oct. 17, there will be a problem session to discuss the problems that you did not have to hand-in. We decided to meet at 7PM, since this time was good for all the people present. Please come to my office, 672 PGH, I will leave a notice with the room number there. Most likely we can use the room where the class is held.

Reading Assignments:

Please read before each class the topics that are going to be covered. What I have in mind is that you read the definitions, the examples, and the statements of the Theorems (we'll discuss the proofs in class).

for Nov. 8: sections 4.9-4.24
for Nov. 6: sections 4.1-4.12
for Nov. 1: sections 3.52 - 3.55, 4.1-4.12
for Oct. 18: sections 3.27 - 3.29, 3.33 - 3.40
for Oct. 16: sections 3.18 - 3.29, 3.33 - 3.35
for Oct. 11: sections 3.7 - 3.19
for Oct. 9: sections 3.5 - 3.14
for Oct. 4: sections 3.1 - 3.4
for the next class: to the end of Chapter 2.
for Sept. 11: sections 2.15 - 2.42.
for Sept. 6: sections 2.15 - 2.30
for Aug. 30: sections 2.1-2.14. Note: the "shortcut" for the proof of Thm. 1.35 (the Schwarz-Cauchy inequality) that I mentioned on Aug. 28 does not work.

The Putnam Competition

The 61^th Putnam Competition (for undergraduates enrolled in universities of SUA and Canada) will be held on December 2, 2000. Please see Prof. Min Ru (minru@math.uh.edu) in PGH 674 for more details.

Midterm 1:

Will be held on Wednesday, Sept.20, 2000. Please bring your student ID.
It will cover all topics discussed by Wednesday, Sept. 13, 2000, that is up to (and including) Thm. 2.33.
It will be an open-book exam. This means that you can bring the textbook (Rudin), course handouts, and your own notes (including anything you might have studied from different books or at different courses). What is forbidden are (xerox copies of) other books, and notes that were not written by you.
You have a (once in a lifetime!) chance of receiving 10 more points for this exam if you turn in by Wed., Oct. 4, the correct (I mean: almost perfect) solutions to the problems for which you did not receive full credit out of #'s 2, 3 and 4 on the exam.

Midterm 2:

Will be held on Wednesday, Oct. 25, 2000. Please bring your student ID.
It will cover topics up to section 3.37 (note that we did not discuss in class the proof of Thm. 3.31).
It will be an open-book exam (same provisions as for the first exam).
If you have questions, I will be in the dept. on Tuesday, Oct. 24, after 2:30PM. This is in addition to my regular office hours.

Midterm 3:

Will be held on Nov. 29, 2000 (Wednesday). Please bring your student ID.
There will be a review session on Tuesday, Nov. 28, at 7 PM.
I will leave a notice on the door of my office with the room number (most likely one near 648 PGH).
It will be an open-book exam (same provisions as for the first exam).
A small typo on the review sheet I handed out: at #15, the problems 11 and 13 are in Chapter 4, not 3.
Check your e-mail for details concerning the "take-home" exam.

Math 4332

This is the home-page for the second semester of the course.

Last modified: April 23, 2001

HOMEWORKS
Chapter	HAND-IN (due date)	Solve only, do not hand-in
1	2, 5 (Aug. 30)	1, 4, 9, 12, 13, 14, 15, 17, 19
2	2, 3, 4 (Sept. 6; assume R is uncountable) 6 (Sept. 13) two problems handed out in class (Sept. 27) 22, 29 (Oct. 4; you have to prove that Q is dense in R)	5, 8, 10, 9, 12, 14, 20, 7, 13 15, 16, 26 (including 23, 24) 21 25
3	1, 2 (Oct. 11) 3, 20 (Oct. 18) 5, 7 (Nov. 1) 13, 14 a, b, c (Nov. 8; extended to Nov. 13)	16 - 18, 21, 22, 24^* - 25 6-14 all problems except #19
4	2, 3 (Nov. 15)	1, 4, 5, 7, 20, 22, 23, 24
5		1, 12, 7, 13, 4 (see Thm. 5.8), 2^, 3, 5, 6, 8^, 9, etc. (use Thm. 5.10)