Math 6320, section 09479 - Fall 2002

Theory of Functions of a Real Variable

• Classes: 2:30-4:00pm TuTh, PGH 350
• Instructor: Andrew Török
• Office: 672 PGH
• Phone: (713) 743-3478
• E-mail: torok@math.uh.edu
• Office hours: see the home-page

Exams

• Exam I: Thursday, Oct. 3
  • The exam will cover all the material discussed by the Thursday before the exam.
  • It will be held during the regular class meeting.
  • Please BRING YOUR STUDENT I.D.

• Exam II: Thursday, Nov. 21
  • The exam will cover all the material discussed by, and including, the Thursday before the exam (Nov. 14).
    That is, up to and including section 5.2.
  • On Tu, Nov. 19, we will continued with the review.
    Please bring your books (so that we don't have to write the problems on the blackboard).
  • Please return the exam on Tuesday, Nov. 26, in class. You can re-do problems that you have already handed in in class.
  • Note that there is HW due next Tuesday as well.

Assignments

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
2.1		1, 2, 3
2.4	15 (due Sept. 10)	8, 9, 10, 23 (use 22, etc.)
2.5		24-36 (37-39 are about the Cantor set)
2.6	47 (due Sept. 12)	40^, 43*, 49, 50 (c^, etc.), 51 (the Cantor function: 48)
1.4		19, 20
2.7		52^, 53, 54^
3.1	1, 2 (due Sept. 12)	3%, 4%
3.2	5 (due Sept. 19)	7%, 8%
3.3	14 (due Sept. 19)   11 (due Sept. 26)	9%, 10%
3.4	15, 16 (due Sept. 26)	this example of a non-measurable set is due to Vitali
3.5	19, 20, 25 (Due Tu, 10/01)	18^, 21^, 24^, 26, 27, 28*
review (Oct. 1)		look at the problems above marked with % (6 problems, these are "short") and ^ (7 problems)
2.7	54 (due Oct. 10)
3.6	29, 30 (due Oct. 17; we proved Prop. 3.24 in class)	32* (explain why  ft   is measurable)
4.1	1 (due Oct. 17)
Oct. 15		1.16-1.18, 1.19, 1.20, 2.36   2.51 (do only: g(x)=h(x) => f cont. at x  ; {h-g < a} is open)   2.53 (see hint given in class; oscf(x)=h(x)-g(x))   3.21, 3.26, 3.27
4.2	parts of 2.51, and 2.53; see details here (deadline extended to Oct. 29) NOTE: there was a typo, corrected on 10/22.	2
4.3	3, 4, 5 (with extra Q: is F uniformly continuous?) (deadline extended to Nov. 7)	6, 7, 9
4.4	14 (include a proof of Thm. 4.17 as well),   19 (use, without proof, problem 2.49f mentioned in 4.18) (due Nov. 12) Hint: both parts of problem 14 can be solved with Thm. 4.17.	all the rest of 10-18
5.1	1 (due Nov. 26)	4
5.2	7 (due Nov. 26)	8-11

NOTE: you have to prove all statements, unless they were discussed in class or are part of the previous homework assignments.

Last modified: December 10, 2002

Course description

Real Analysis by H. L. Royden (publisher: Prentice Hall)

The prerequisites are MATH 4332 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)

Last modified: August 29, 2002