INFORMATION FOR MATH 5453-001
Real Analysis I
Fall 2009
(Last updated on 31 August 2009)
Text: Real Analysis (3rd ed), by H. L. Royden, Macmillan, 1988, ISBN 0-02-404151-3 (I believe this textbook is now just published in a paperback format).
Syllabus and course objectives: This course is intended to provide a rigorous mathematical introduction to the theory of the Lebesgue integral. As secondary goals, the course will delve into some of the more subtle aspects of the real number system (such as completeness and category) and the rudiments of metric spaces and infinite dimensional vector spaces (as preparation for more advanced courses such as functional analysis). Topics to be covered will include: the real number system, metric spaces, and notions of completeness and Baire category; the Baire Category Theorem and applications; a rapid review of the Riemann integral, including its basic properties and limitations; fields of sets, σ-algebras of sets, and measures; construction of measures via the Caratheodory extension theorem (with Lebesgue measure on the real line as the preeminent example); measurable functions, their definition and basic properties; the construction of the abstract Lebesgue integral and its basic properties; the monotone convergence theorem, Fatou's lemma, the dominated convergence theorem, and applications; the connection between the Lebesgue and Riemann integrals; differentiation, functions of bounded variation, absolutely continuous functions, and the "fundamental theorem of calculus" in the context of the Lebesgue integral for real-valued functions of a real variable.
Prerequisites: MATH 4433 (or an equivalent introductory course in analysis). In terms of specific prerequisites by topic, the main one is some familiarity with the reading and construction of mathematical proofs (sometimes called "mathematical maturity"). It will also be assumed that your undergraduate analysis course has introduced you to such topics as the least upper bound axiom, sequences, limits, continuity, the basic topology of the real line (open, closed, compact, and connected sets), derivatives and the mean-value theorem, and the rigorous definition of the Riemann integral. We may rapidly review some of these topics, but some prior exposure to them is probably essential for your success in this course.
Grading: Your grade will be determined by your performance on the following course work:
Attendance: You are required to attend class on those days when an examination is being given; attendance during other class periods is also strongly encouraged. Be advised that you are fully responsible for the material covered in each class, whether or not you attend. Make-ups for missed exams will be given only if there is a compelling reason for the absence , which I know about beforehand and can document independently of your testimony (for example, via a note or phone call from a doctor, parent, or clergyman).
Homework: Homework will be assigned regularly throughout the semester in the form of do-at-home problem sets (I anticipate the frequency of problem sets to be once every two weeks or so). With a few exceptions, most of the problem sets will be collected for grading and will have a specified due date, in which case they must be turned in by no later than 3:30PM of the due date. I will not accept late papers unless you have made prior arrangements with me for turning in a late assignment. Problem sets will be posted on the homework and exam information page for this course at least one week before the assignment is due. After a homework assignment has been collected, solutions to the problems comprising that assignment will be posted on the aforementioned homework and exam information page.
Note #1: You should view the assigned problem sets as containing the bare minimum number of problems required to attain a basic level of mastery of the material. Depending on your own proclivities, you may need to work additional problems to achieve a higher level of mastery of the material (say, for example, a level of mastery sufficient to pass the PhD Qualifying Exam).
Note #2: I deem it acceptable for students to work in groups and/or with a tutor as they make their preliminary efforts to explore and work through homework problems. However, after any such preliminary and cooperative efforts, I expect you to write up your final homework papers individually and without outside assistance. The acts of simply copying another student's homework paper, or writing a problem solution as dictated by a tutor or other helper, will constitute academic misconduct and will be prosecuted according to the University's Academic Misconduct Code (see below).
Some Important Dates :
Policy on W/I Grades : These grades are frequently misunderstood by students, so please read this section carefully. Through Friday, October 2, you can withdraw from the course with an automatic W. In addition, it is my policy to give any student a W grade, regardless of his/her performance in the course, through the extended drop period that ends on Friday, October 30. However, after October 30, you can only drop via petition to the Dean of your college. Such petitions are not often granted. Furthermore, even if the petition is granted, I will give you a grade of "Withdrawn Failing" if you are indeed failing at the time of your petition. Thus it is in your own best interest to drop the course on or before October 30 if you think there is a reasonable chance that you will not want to see the course through to the end.
The grade of I (Incomplete) is not intended to serve as a benign substitute for the grade of F. I will only give an Incomplete if a student has completed the majority of the work in the course (for example everything except the final exam), the course work cannot be completed because of compelling and verifiable problem beyond the student's control, and the student expresses a clear intention of making up the missed work as soon as possible. Note also that the Mathematics Department requires that instructors and students execute and sign a written Incomplete Contract before a grade of Incomplete can be given.
Academic Misconduct: All cases of suspected academic misconduct will be referred to the Dean of the Graduate College for prosecution under the University's Academic Misconduct Code. The penalties can be quite severe. Don't do it! For more details on the University's policies concerning academic misconduct click here . For information on your rights to appeal charges of academic misconduct click here . Students are also bound by the provisions of the OU Student Code, which can be found here .
Students With Disabilities: The University of Oklahoma is committed to providing reasonable accommodation for all students with disabilities. Students with disabilities who require accommodations in this course are requested to speak with the instructor as early in the semester as possible. Students with disabilities must be registered with the Office of Disability Services prior to receiving accommodations in this course. The Office of Disability Services is located in Goddard Health Center, Suite 166: phone 405-325-3852 or TDD (only) 405-325-4173.