Math 5503: Real Variables I

Download PDF version of this syllabus.
Homework, past exams.
My notes from Reals I and II, U. of Illinois, 1971-72.

Instructor: Daniel H. Luecking

Office: SCEN 354   Telephone: 575-6327

Office Hours: 10:45-11:35 and 1:30-2:30 MWF.

Text: Real Analysis by H.L. Royden, 3rd edition.

Material to be covered:

Chapter 1. Set theory. 4 days

You are responsible for all sections of this chapter, but we will cover only sections 4, 5 and 8 in class, along with a few other selected topics, including metric spaces.

Chapter 2. Convergence and continuity. 5 days

You are responsible for all sections in this chapter. I will cover only some of the topics, but I will include metric spaces in the discussion (see chapter 7).

Chapter 3. Lebesgue measure. 7 days.

Section 1. Introduction.     Section 2. Outer measure.     Section 3. Measurable sets and Lebesgue measure     Section 4. A nonmeasurable set.     Section 5. Measurable functions: sums, products, composition and limits.     Section 6. Littlewoods three principles, Egoroff's Theorem and Lusin's Theorem .

Chapter 4. Lebesgue integration. 7 days.

Section 1. The Riemann integral.     Section 2. The Lebesgue integral of a bounded function on a set of finite measure.     Section 3. The integral of a nonnegative function.     Section 4. The general Lebesgue integral.     Section 5. Convergence in measure.

Chapter 5. Differentiation and integration. 7 days.

Section 1. Monotone functions.     Section 2. Functions of bounded variation.     Section 3. Derivative of an integral and vice versa.     Section 4. Absolute continuity.     Section 5. Convex functions.

Chapter 6. Lp spaces: completeness and approximation. 7 days.

Section 1. Normed spaces and Banach spaces.     Section 2. Minkowski and Hölder inequalities.     Section 3. Convergence and completeness.     Section 4. Approximation and separability.     Section 5. The Riesz Representation Theorem.

Other topics. 5 days.

Various topics as time permits.

Other Information

Exams and homework: There will be a midterm exam some time after Chapter 3 and a final exam on Wednesday, Dec. 13, 10:15am-12:15pm. Homework may consist of problems from the book or problems of my own devising. Homework will probably be assigned every two weeks or so. I will never assign homework before the previous one is graded.

Grading: The midterm exam will be worth 200 points, the final exam will be worth 300 points. Homework sets may be worth a variable number of points, but the total at the end of the semester will be rescaled so as to be worth 200 points. All of these scores (midterm, final, and homework total) may be curved if I think it necessary. Grades will be assigned on the following scale: 595-700 = A, 455-594 = B, 350-454 = C; 280-349 = D.

Policy on Class Cancellation: If weather seems to threaten cancellation of class, check with the University: if it is open, I will have class. If this class is cancelled for any other reason, the Math office should be able to inform you (check in room 309). If I fail to show up by 9:45, one of you should come to my office and get me.

Statement on Academic Honesty: Submitting the work of another as your own is a serious violation of the University's policy on academic honesty, and will result in disciplinary proceedings when detected. This includes any take-home assignments ! Do not accept or solicit help from anyone except me and do not offer help to anyone on them. If you are uncertain what constitutes a violation, talk to me before you do it. If you are in violation you will receive no points on the entire assignment affected. In the worst case, you could fail the course and be dismissed from the University.

Quoting the University: Each University of Arkansas student is required to be familiar with and abide by the University's `Academic Integrity Policy' which may be found at Students with questions about how these policies apply to a particular course or assignment should immediately contact their instructor.

Other Information: The midterm will be taken during one class period. Whatever you fail to complete can be taken home and turned in later. Problems turned in later will be counted half of what is completed in class.

The final exam will have two parts. The take-home part (100 points) will handed out on the last day of class and is to be turned in at the beginning of the final exam period. The sit-down part will be worth 200 points.

File translated from TEX by TTH, version 3.85.
On 09 Aug 2017, 15:06.