MATH 323: Real Analysis

Fall 2006

Catalog Description: An axiomatic treatment of the topology of the real line, real analysis, and calculus. Topics include neighborhoods, compactness, limits, continuity, differentiation, Riemann integration, and uniform convergence.

Informal  Description:   The calculus that you learned in 121 and 122 is very much in the spirit of what Newton and Leibniz formulated near the end of the 1600's. Specifically, the concepts of differentiation and integration are built on a rather imprecise understanding of limits. But by the 1800's, the expanding ideas of calculus -- most notably in the direction of infinite series -- caused a serious crisis as it became apparent that the intuitive definitions were insufficient for resolving some sticky convergence questions. Cauchy, Dirichlet, Weierstrass, et al., managed with great effort to put the theory of calculus on a rigorous foundation, and in so doing were able to extend its power to new and exciting areas. Some versions of this course include an exploration of a few of these areas (e.g. Fourier series), while others return to the very beginning and actually construct the real numbers on which the theory of limits depends.

         MATH 323 is a required course for mathematics majors.  It puts some of the core material of Calculus I on a firm and rigorous footing.  It also reveals some fascinating mathematics that is not accessible in first-year calculus. MATH 323 provides a foundation for one of the major branches of mathematics:  mathematical analysis.  It is important material for further study of calculus, complex variables, probability, mathematical statistics, econometrics, engineering mathematics, mathematical modeling, topology, numerical analysis, and differential equations.

Other Course Objectives:

• To learn how to write extended, rigorous mathematical proofs.

á      To appreciate why an intuitive understanding of calculus (learned in MATH 121 and 122) is insufficient as a foundation for building more complicated and powerful mathematical tools.

á      To enjoy the payoff from analytical rigor by learning some newly accessible topics. Some of these will be surprises and quite provocative!

Prerequisites:      Multivariable Calculus equivalent to MATH 223 (Multivariable Calulus)

Distribution Requirements:    Completion of this course will satisfy the DED (deductive reasoning and analytical processes) distribution requirement.

Instructor: Michael Olinick, 314 Warner, telephone extension: 5559. Home Phone: 388-4290. Office  Hours: Monday, Wednesday and Friday from 9:30 to Noon, Tuesday and Thursday from 11 AM to 1 PM.  I have another class that meet on Tuesday and Thursday mornings from 9:30 to 10:45  so I am not available then. I would be happy to make an appointment to see you at other  mutually convenient times.

Meeting Times:      Tuesday and Thursday from  1:30 to 2:45 PM  in Ross Commons Dining RCD B11

Textbook: 

            Stephen Abbott, Understanding Analysis, New York: Springer.

You can find more information about the text, including reviews and a list of typographical errors at

http://community.middlebury.edu/~abbott/UA/UA.html

            I have placed a copy the book on reserve in the College Library. There may be other readings in the course that will be on reserve.

Requirements: There will be daily written homework assignments and numerous student presentations. See schedule for due dates.