Math 6321, section 09878 - Spring 2003

Theory of Functions of a Real Variable, part II

Classes: 2:30-4:00pm TuTh, 202 AH
Instructor: Andrew Török
Office: 672 PGH
Phone: (713) 743-3478
E-mail: torok@math.uh.edu
Office hours: see the home-page

Teacher Evaluations

The Math Dept will collect this semester the evaluations via an online form.
See http://online.math.uh.edu.

Exams

Exam I: will be a take-home exam. You will receive it on Feb. 20.
Exam II: will be a take-home exam. You will receive it in class on Tuesday, April 22, and will be due in class on Thursday, April 24.
The Final Exam: will be a take-home exam.
See e-mail for major changes.
I will post it on the web, and you will have three days to finish it. You will have to turn it in by 5PM, at my office, on Tuesday, May 6.

Assignments

Statistics

The best five HW's will count for 100 points.
Taking the best five scores of the first six homeworks for each student, the class average is 69.8%.
You can turn in more HW's (see those marked "optional" in the table); I will take into account the best five scores.

Problems

The problems below are from Royden's book. Please write the solutions on separate pages (i.e., do not hand in a note-book).

Some of the problems might have the answer or a hint in the textbook. If you have to hand in such a problem, you are welcome to use the hint but have to still write a complete solution.
More difficult problems are marked by a ^*.

HOMEWORKS

Section HAND IN (due date) Solve only, do not hand in

5.4 12, 20 (due Jan. 30; extended to Feb. 6)
Note that at 20(c) one has to add the assumption that f is continuous. 13, 14, 18, 21, 22; see also 3.28 (page 71)

5.5 26 (due Feb. 6) 23-28

6.1 2 (due Feb. 6) 1, 3, 4

6.3 10, 11, 13, 14, 15, 16, 17, 18

6.4 (due Feb. 13, exteded to Feb. 18) 19, and
"Prove that C([0,1]) is not dense in L^infinity([0,1]). What is the closure of C([0,1]) in L^infinity([0,1])?" (See explanation below)

6.5 (due Feb. 20) 21, and: Show that

l¹(N) = l^one(N) is contained in l²(N).
the linear map T:l¹(N) --> l²(N), T(x)=x IS bounded (sorry, this was an error; the map is bounded).
l²(N) is NOT contained in l¹(N).
(This is what I mentioned in class on 2/11; the same applies to all p < q.) 22, 23

11.5 (due March 20) 27, 28 30, 31, 32

11.6 (due April 1) 34, 35 36, 37, 39, 38 (this gives both 11.23 and 11.24 in one stroke - although it is not clear from the problem - see the proof of Radon-Nikodym in Rudin, Real and Complex Analysis; hope to discuss Prop. 10.28 later )

11.7 42, 43, 44, 46

for Tu, 4/1 read sections 12.1 and 12.2, up to (and without) Prop. 12.7.

12.2 (due April 22; OPTIONAL) 4, 5 3, 4, 5, 6, 7, 10, 11 (we did most of these for the Lebesgue measure)

12.5 (due April 22; OPTIONAL) 22, 24 19, 20, 21, 22, 24, 25, 26^*, 27 (do at least the uniqueness part), 31 (another defn of the integral)

10.1 1, 3

10.2 13, 14

10.3 17, 20

10.4 26, 28 (use Prop 10.11 for b), 29 (there is a typo in the hint: f is defined on Y)

By closure in L^infinity, we mean that the norm used is the L^infinity norm. Hence, f_n -> f if the essential sup of (f_n-f) goes to zero as n -> infinity. [In the HW problem, we want to look at the limits one can obtain if f_n are continuous.]

HOMEWORKS
5.4	12, 20 (due Jan. 30; extended to Feb. 6) Note that at 20(c) one has to add the assumption that f is continuous.	13, 14, 18, 21, 22; see also 3.28 (page 71)
5.5	26 (due Feb. 6)	23-28
6.1	2 (due Feb. 6)	1, 3, 4
6.3		10, 11, 13, 14, 15, 16, 17, 18
6.4	(due Feb. 13, exteded to Feb. 18) 19, and "Prove that C([0,1]) is not dense in L^infinity([0,1]). What is the closure of C([0,1]) in L^infinity([0,1])?" (See explanation below)
6.5	(due Feb. 20) 21, and: Show that l¹(N) = l^one(N) is contained in l²(N). the linear map T:l¹(N) --> l²(N), T(x)=x IS bounded (sorry, this was an error; the map is bounded). l²(N) is NOT contained in l¹(N). (This is what I mentioned in class on 2/11; the same applies to all p < q.)	22, 23
11.5	(due March 20) 27, 28	30, 31, 32
11.6	(due April 1) 34, 35	36, 37, 39, 38 (this gives both 11.23 and 11.24 in one stroke - although it is not clear from the problem - see the proof of Radon-Nikodym in Rudin, Real and Complex Analysis; hope to discuss Prop. 10.28 later )
11.7		42, 43, 44, 46
for Tu, 4/1	read sections 12.1 and 12.2, up to (and without) Prop. 12.7.
12.2	(due April 22; OPTIONAL) 4, 5	3, 4, 5, 6, 7, 10, 11 (we did most of these for the Lebesgue measure)
12.5	(due April 22; OPTIONAL) 22, 24	19, 20, 21, 22, 24, 25, 26^*, 27 (do at least the uniqueness part), 31 (another defn of the integral)
10.1		1, 3
10.2		13, 14
10.3		17, 20
10.4		26, 28 (use Prop 10.11 for b), 29 (there is a typo in the hint: f is defined on Y)

NOTE: you have to prove all statements, unless they were discussed in class or are part of the previous homework/exam assignments (in either this course or Math 6320 last Fall). This also applies to statements made in the exercises of the book.

Last modified: May 5, 2003

Course description

This is the continuation of Math 6320 taught in Fall 2002. The book that we will be mainly using this semester is

Real Analysis

In Fall 2002 we covered most of the first 5 chapters.

The prerequisites are MATH 6320 or consent of the instructor.

The course will introduce Lebesgue measure and integration, differentiation of real functions, functions of bounded variation, absolute continuity, the classical L^p spaces, general measure theory, and elementary topics in functional analysis.

There will be regular homework assignments (the lowest grade will be dropped), and the exams (some might be take-home).

Other books that might be useful are (library call numbers included):

Frank Jones: Lebesgue integration on Euclidean space, QA312.J58 1993 (it is more elementary than Royden; deals only with measure on Rⁿ)
M. E. Munroe: Measure and integration, UHCL QA312.M74 1971, 1st edition QA251.M8 1953 (this is an old book; it was used a few years ago for this course)
Avner Friedman: Foundations of Modern Analysis, QA299.8.F74 1982
Walter Rudin: Real and complex analysis, QA300.R82 (use the "real" part only; this is nice and concise - but maybe harder than Royden)
G. B. Folland: Real analysis; modern techniques and their applications, QA300 .F67 1999 (this is a quite complete book, but not an easy one)

These books were on reserve during the Fall semester. I will keep them on reserve.

Last modified: January 13, 2003

HOMEWORKS
Section	HAND IN (due date)	Solve only, do not hand in
5.4	12, 20 (due Jan. 30; extended to Feb. 6) Note that at 20(c) one has to add the assumption that f is continuous.	13, 14, 18, 21, 22; see also 3.28 (page 71)
5.5	26 (due Feb. 6)	23-28
6.1	2 (due Feb. 6)	1, 3, 4
6.3		10, 11, 13, 14, 15, 16, 17, 18
6.4	(due Feb. 13, exteded to Feb. 18) 19, and "Prove that C([0,1]) is not dense in L^infinity([0,1]). What is the closure of C([0,1]) in L^infinity([0,1])?" (See explanation below)
6.5	(due Feb. 20) 21, and: Show that l¹(N) = l^one(N) is contained in l²(N). the linear map T:l¹(N) --> l²(N), T(x)=x IS bounded (sorry, this was an error; the map is bounded). l²(N) is NOT contained in l¹(N). (This is what I mentioned in class on 2/11; the same applies to all p < q.)	22, 23
11.5	(due March 20) 27, 28	30, 31, 32
11.6	(due April 1) 34, 35	36, 37, 39, 38 (this gives both 11.23 and 11.24 in one stroke - although it is not clear from the problem - see the proof of Radon-Nikodym in Rudin, Real and Complex Analysis; hope to discuss Prop. 10.28 later )
11.7		42, 43, 44, 46
for Tu, 4/1	read sections 12.1 and 12.2, up to (and without) Prop. 12.7.
12.2	(due April 22; OPTIONAL) 4, 5	3, 4, 5, 6, 7, 10, 11 (we did most of these for the Lebesgue measure)
12.5	(due April 22; OPTIONAL) 22, 24	19, 20, 21, 22, 24, 25, 26^*, 27 (do at least the uniqueness part), 31 (another defn of the integral)
10.1		1, 3
10.2		13, 14
10.3		17, 20
10.4		26, 28 (use Prop 10.11 for b), 29 (there is a typo in the hint: f is defined on Y)