Math 536 syllabus

Real Analysis II

Bulletin description:

A continuation of MA 535. Topics covered include sequences and series of functions, differentiation and integration in several variables, an introduction to to the Lebesgue integral and differential forms as time allows. Prerequisite: MA 535, or consent of instructor.

Textbook:   Walter Rudin, Principles of Mathematical Analysis (3rd ed.), McGraw-Hill, Inc. (New York), 1976,  
ISBN–13: 978–0070542358 or a text at a similar level chosen by the instructor.

Coverage:   Chapters 7, 8, 9, 11 and possibly 10 of Rudin’s text.
Course work:   Graded homework is an important part of the course. Written in-class exams are encouraged, since masters students must take a written qualifying exam in Real Analysis.

Learning Objectives:

On successful completion of the course, students will understand the implications of uniform convergence for sequences and series of functions, and know and be able to apply the Stone- Weierstrass theorem. They will know the basic theory of power series and Fourier series, including the series definition of the complex exponential function, criteria for convergence and Parseval’s Theorem, and be able to compute and work with these series expansions. They will understand differentiability of functions of several variables, and know and be able to use the inverse function theorem and implicit function theorem. They will know the basics of Lebesgue measure and integration and
Lp space. They will be familiar with at least one advanced approach to integration of functions of several variables (via differential forms, Riemann sums on Jordan regions or Lebesgue measure.) Students should be adept at constructing proofs of basic results in analysis by the end of this course.

Updated:   September 23, 2014