Syllabus for MAT 7600 \ Section 001 \ Call 15403
Fall 2007 (MWF 08:30--09:25) 0111 State

MAT 7600 Real Analysis I. Prerequisites: MAT 5610 or consent of instructor. Lebesgue measure; general measures; measurable functions; integration (monotone and dominated convergence theorems); function spaces; Lebesgue spaces; modes of convergence; product measures; Fubini theorem.

MAT 7610 Real Analysis II. Prerequisites: MAT 7600 or consent of instructor. Differentiation; relationship between differentiation and integration; Radon-Nikodym theorem; Fourier transforms; Hilbert and Banach spaces; selected topics.

Overview. This real analysis course is mainly reserved for mathematicians. Perhaps, the material included and the way of teaching is particular to each instructor, but all have a common list of topics as above. Several books (not just one!) are our textbooks, but you will have the lecture notes (in pdf) of our course. You should be more of less familiar with this background material, we are going to cover it very quickly, please read it in advance and ask questions in class. Frequently, check here for an update (course outline) on this syllabus.

Attendance to class will be excused (only) in the case of sickness or other emergency and you will be asked to document the circumstances which caused you to miss the lecture. Ask questions! It's your responsibility to ask about anything you don't understand. Write down the things that bother you while you're reading the text or working on problems, so you'll be ready with a list of questions when you come to class and/or office hours. There's no such thing as a stupid question, usually other students are grateful that you asked the question. Cell Phones and Pages should be turned off during class.

Homework Assignments will be given continuously. Keeping up with the homework is probably the single most important things you can do to improve your chances for a good grade. Clarity and brevity will raise your grade, while unclear explanations and unnecessary material will lower it. The discussion should be typed, printed, or legibly hand-written on ordinary size paper. Illegally documents will not be graded. Essentially, you will have to learn how to write and how to present (verbal discussion in class and/or in my office) mathematics. The main homework assignment is to read the lecture notes, and to confront the references.

Grading. Homework Assignments will give up to 40 points. A Midterm Examination (Mon 05/Nov/07) and Final Examination (Fri 14/Dec/07, 08:00 - 10:30 a.m.) will give 30 points each, to complete the 100 points of the grade. The final may include a short oral discussion or presentation.

Some Books.

[1] H. Bauer, Measure and Integration Theory, Walter De Gruyter Inc, Berlin, 2001.
[2] E. DiBenedetto, Real Analysis, Birkhauser Boston Inc., New York, 2002.
[3] J.H. Dshalalow, Real Analysis, Chapman & Hall / CRC Press, Boca Raton (FL), 2001.
[4] R.M. Dudley, Real Analysis and Probability Cambridge University Press, Cambridge, 2002.
[5] W. Filter and K. Weber, Integration Theory, Chapman & Hall, London, 1997.
[6] E. Hewitt and K. Stromberg, Real and Abstract Analysis, Springer-Verlag, New York, 1965.
[7] I.K. Rana, An Introduction to Measure and Integration, Amer. Math. Soc., Providence (RI), 1997.
[8] H.L. Royden, Real Analysis, Prentice-Hall, Englewood Cliffs (NJ), third printing, 1988.
[9] W. Rudin, Real and Complex Analysis, McGraw-Hill Book Co., New York, third edition, 1987.
[10] M.E. Taylor, Measure Theory and Integration, Amer. Math. Soc., Providence (RI), 2006.
[11] R. Wheeden and A. Zygmund, Measure and Integral, Marcel Dekker, New York, 1977.

The books Hewitt-Stromberg[6], Royden[8], Rudin[9] and Wheeden-Zygmund[11] has been successfully used as text in previous years, and certainly they are recommended. Hewitt-Stromberg[6] is very formal in many aspects, Rudin[9] gives more emphasis to the complex analysis and Wheeden-Zygmund[11] is more real harmonic analysis, while the material in Chapters 1 and 2, and most of Chapters 7, 8 and 9 of Royden[8] are supposed known for MAT-7600. There are a lot of exercises in any of them. The book by Filter-Weber[5] presents ``Measure and Integration Theory'' using lattices and Daniell's approach, is a shorter version of Hewitt-Stromberg[6] in some sense, with more details in other aspects. The book by Dshalalow[3] is easier to read, Part I is assumed known for MAT-7600, where we essentially cover the remaining parts of the book. There are many other books that can be used in our course, e.g., the books by Bauer[1], Rana[7] and Taylor[10] are excellent texts for MAT-7600.
DiBenedetto[2] and Dudley[4] are two books that you may want to have in you library, they are extremely well written, with many exercises and a great diversity of material. Anyway, it is a very good practice to read the reviews on any of the above book at MathSciNet. The books (in pdf) General Topology (by J.M. Moller) and Topology Without Tears (by S.A. Morris) may be of your interest.
All of the above books (and many more) are textbooks for our course MAT-7600 and MAT-7610, and in a way, a little of each of them is involved. Nevertheless, each student will have the lecture notes of all the subjects covered during the course, but exercises are taken of the above sources.
At this link you will find the Lecture Notes in pdf, which are periodically updated as the course evolves.

Office Hours. Mon and Wed 10:35 -- 11:30, and by appointment. FAB 1229, Phone (313) 577-3196, Dr. J.L. Menaldi (e-mail: menaldi@wayne.edu), 30/Aug/07.

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