MATH 4331-4332, Introduction to real analysis

This page last updated on September 4, 2009.

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.) If you are interested in joining the class but are not sure about your background and/or you wish to do some preliminary summer reading, please let me know. I am in the process of putting together some summer reading. This will be not be extensive, but if you do the reading it should make taking the class much easier.

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.

PREREQUISITES: The information on the department website is a little out-of-date. MATH 3334 is NOT a prequisite for the MATH 4331/2 sequence. It is not even particularly useful for MATH 4331, though it might well be of some use for MATH 4332. 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 - or reading through the first chapter of my course notes. Reading through most of chapter 1 before the course starts is strongly advised.
Keep an eye on these pages as, as some stage over the summer, I will list a few topics that you should be familiar with before starting the class. One of the reasons I am working on developing these preliminary reading topics is that in recent years the class has been quite diverse in background; ranging from the extremely well prepared to the not at all well prepared. It helps if everyone has at least a working knowing of a few core topics (eg: basic set theory and notation, ratio test).