University of Houston
Department of Mathematics
MATH 4331; 4332, Introduction to Real Analysis

Prerequisites: MATH 3333 or consent of instructor.

Textbook: There is no set text book - a set of notes (PDF file) will be provided during the summer or at the start of the semester. there will be lecture notes available by email - if you want them before the class starts, you will need to be enrolled in the class and then email me at mikefield@gmail.com requesting a copy of the notes. They should be available by the beginning of August. A good reference book - especially if you want to do some preliminary reading is Kaplansky, Metric spaces (Approximately $23 from Amazon, there could well be second hand copies in the UH bookshop.)

Course Description:

Lectures are likely to be on Monday and Wednesday, 4-5:30. Location: TBA.
Grading and examination policy will be detailed in Lecture 1. (For Spring 2009 see here).

The emphasis in MATH 4331 will be on 1-variable theory and results from "classical analysis". Topics covered will include infinite series, sequences, functions (continuous, analytic, smooth), uniform convergence, Weierstrass Approximation theorem, Fourier series, the Gamma-function and the Euler-Maclaurin formula. There will also be a fairly extensive introduction about the real number system.

Semester 2 (MATH 4332) will develop metric spaces and include applications of the contraction mapping lemma. Topics covered will include metric spaces, open and closed sets, interior, closure, limit points, continuity, sequential compactness, and completeness. There will also be several lectures on fractals and iterated function systems as well as some differential calculus on RN (self contained, MATH 3334 not essential; though it will not hurt having done it).

One big difference from the syllabus prior to 2008 will be the order of material. The focus of the course will be to end your senior year with a bang rather than a whimper. In particular, I will avoid ending the course with unmotivated and dull technical material that amounts to little more than preparation for graduate courses. We will see the applications - they will not be deferred to later.

PREREQUISITE NOTE: MATH 3333 is strongly advised. If you have not done MATH 3333, but have done courses beyond Calculus I-III (for example, MATH 3363) then you may be able to take the MATH 4331/2 sequence. However, going into the MATH 4331/2 sequence immediately after doing the calculus sequence is not advised. If you are in doubt, email me at mikefield@gmail.com and let me know what maths courses you have already taken and with what grades. Even if you have taken MATH 3333, I advise reading the introductory chapter(s) of Kaplansky over the summer - they are not long and not hard and they give some of the flavor of what we will be doing in both the first and second semesters.