MTHU550, Analysis, Spring 2008: Syllabus

Text: Real Analysis --- A Constructive Approach, by Mark Bridger (John Wiley, 2006).

Instructor: Prof. Mark Bridger, 527 NI

E-mail: bridger@neu.edu

Website: www.math.neu.edu/~bridger/U550/U550.htm

Office Hours: To Be Announced

Course Outline: Real analysis is the theoretical underpinnings of the calculus. We begin by studying how the integers and rationals are constructed from the whole numbers (Chapter 0 of the book). We then move on to examine the more difficult construction of the real numbers from the rationals (Chapter 1). Traditionally this has been done using Dedekind “cuts”, Cauchy sequences, or nested intervals. We will adopt a newer approach using “interval arithmetic” and families of intersecting and arbitrarily small rational intervals. We prove that the reals, as constructed, are complete.

Next (Chapter 2), we prove an Inverse Function Theorem, using certain inequalities called Lipschitz conditions as hypotheses. We apply this to deduce the existence of nth roots, then the exponential function, and finally logarithms.

Instead of studying pointwise continuity and differentiability, we introduce the corresponding uniform notions (Chapters 4 and 6), and show that all of the usual functions of calculus are uniformly continuous and differentiable on bounded intervals. We prove a “mean value” type inequality, the Law of Bounded Change. In between these two topics, we use the completeness of the reals to prove that uniformly continuous functions have Riemann integrals (Chapter 4). Finally, we deduce the Fundamental Theorem of Calculus and, as time permits, give further applications to other theorems of calculus (Chapter 5).

Goals: In addition to covering the topics listed above, the other goals of MTHU550 are: (a) becoming acquainted with the nature of modern mathematical abstraction and (b) learning

how to write careful, logical, and understandable mathematical proofs.

Written Work and Grades: There will be frequent short quizzes, at least one hour-exam, and a final exam, totaling 50% of the course grade. There will be approximately half a dozen problem sets for the remaining 50% of the grade. From time to time problems and hints will be posted at my website (see above).

Note: The problem sets are a very important part of this class; some of them you may find challenging. Do not put them off till the last minute: they take time and care. You may work with others in the course -- in fact, you are encouraged to do so. However, everyone is expected to hand in individual write-ups of the work. If you need additional help, don't hesitate either to come to office hours or e-mail me (I check my mail several times a day, and will respond as soon as I can to any questions).

Attendance: Attendance will be taken; you are allowed a maximum of 3 absences, period!

If you are having trouble with this course, or anticipate any problems, please see me during office hours as soon as possible. If you become ill, you are responsible for the work you missed, so notify me and the others in your study group.