Math 515, Section A
Real Analysis I
Fritz Keinert
Fall 2009

Instructor

Fritz Keinert
464 Carver
294-5223
keinert@iastate.edu
Office hours: M 9-11am, 1-2pm, WF 10-11am,
or by appointment.
Class meetings: MWF 11:00-11:50am, Carver 290

Current Scores

Current scores can be found in WebCT.

Homeworks

Homework 1, due Friday, Sep 18, 2009. Solutions can be found in WebCT.

Homework 2, due Friday, Oct 2, 2009. Solutions can be found in WebCT.

Homework 3, due Friday, Oct 16, 2009.

Homework 4, due Friday, Nov 6, 2009.

Homework 5, due Friday, Nov 20, 2009.

About the Course

Textbook: H. L. Royden Introduction to Real Analysis 3rd edition Macmillan ISBN 0-02-404151-3

This course is the first part of the two-part series Math 515/516, but many students only take the first semester.

Catalog Description of Math 515: Measure and integration with special emphasis on Lebesgue measure, modes of convergence of sequences of functions, decomposition of measures, differentiation, metric spaces, L^p spaces.

This course is also the basis for the real part of the Real and Complex Analysis Qualifying Exam. The following topics are listed as being allowed on the exam, so we need to make sure to touch on them.

1. Lebesgue measurable sets and functions in one dimension: σ-algebras, outer measure, Borel sets, Cantor set, Egorov's theorem
2. Lebesgue integration theory in one dimension: approximation by simple functions, Fatou's lemma, Monotone and Dominated Convergence Theorems, comparison with Riemann integral
3. One variable differentiation theory: differentiation of monotone functions, functions of bounded variation, absolute continuity of functions, Fundamental Theorem of Calculus, Lebesgue set, convexity
4. Product integration: Fubini and Tonelli theorems, Lebesgue measure and integral in Rⁿ
5. L^p spaces, Jensen, Hölder and Minkowski inequalities, l^p spaces, dual spaces, Riesz representation theorem, density and approximation theorems, special case of L²

Catalog Description of Math 516: Continuation of Math 515. Hilbert and Banach spaces, product integration, Fubini's theorem, other topics at the discretion of the instructor.

Obviously the catalog and the list of exam topics contradict each other. I am planning to pay more attention to the list of qualifying exam topics.

Topics in Math 515: Royden chapters 1-6, parts of chapters 7 and 10, plus extra material on product integration and Fubini's theorem (which is not in the book).

Topics in Math 516: Royden chapters 11 and 12 (general measure theory), plus material from other sources on more Banach and Hilbert space theory, distributions and the Fourier transform (probably based on R. Strichartz, A Guide to Distribution Theory and Fourier Transforms, World Scientific 2003).

General Organization of the Class

There will be two sections (A and B) of Math 515, taught by myself and by Prof. Yiu Tung Poon, at the same hour. Math 515 is a fundamental course for all Math graduate students, and students expect detailed feedback on their homework. The department has split the course into two to distribute the grading load. We will discuss this further at the first class meeting. Home page for Poon's section.

For future reference: the other section meets in room 128. If I should ever be sick or not show up for some other reason, go to room 128 and attend the other section.

We plan to have common homework assignments and a common final exam for both sections, and to also share the grading.

Tentative outline of the semester:

Chapters 1, 2: 3 weeks, HW 1 (Aug 24-Sep 11)

Chapter 3: 2 weeks, HW 2 (Sep 14-25)

Chapter 4: 2 weeks, HW 3 (Sep 28-Oct 9)

Extra material on product integration, Fubini's theorem: 1 week (Oct 12-16)

Chapter 5: 2 weeks, HW 4 (Oct 19-30)

Chapter 6: 2 weeks, HW 5 (Nov 2-13)

Chapters 7, 10: 3 weeks, HW 6 (Nov 16-Dec 11)

Final Exam (Wednesday, Dec 16, 9:45-11:45 am)

Students With Disabilities

If you have a documented disability and require accommodations, you should obtain a Student Academic Accommodation Request (SAAR) from the Disability Resources office (Student Services Building, Room 1076, 294-6624 or TDD 294-6335, disabilityresources@iastate.edu or accommodations@iastate.edu). Please contact your instructor early in the semester so that your learning needs may be appropriately met.

More information about disability resources in the Mathematics Department can be found at http://www.math.iastate.edu/About/AccommodationPol.html.