Real Analysis I

Stony Brook

Fall 2009

TuTh 11:20am-12:40pm Mathematics 4--130

Professor: Mikhail Movshev
Office hours: Mon 12:00-2:00

Grader:Young Nam
e-mail ynam ad math dot sunysb dot edu

Textbook: A First Graduate Course in Real Analysis, by Daryl Geller. You can get it at P-143 from the Graduate Secretary ($40)
Additinal textbookReal Analysis: Modern Techniques and Their Applications (2nd edition) by Gerald B. Folland
Final Exam: Tuesday, December 15, 2:15PM-4:45PM Mathematics 4--130

Grading: 50% Final Exam, 25% Midterm, 25% Homework.
Homework will be assigned every Tuesday in class (starting the second week), and be due the following Tuesday. Homework assignments will usually be available at this website by Monday evening. No late homework will be counted towards the final grade.

This is the first semester of the three-semester sequence of graduate-level real analysis offered at Stony Brook. This semester is divided into two parts.

The first part consists in turn of:

  1. Review of rigorous foundations of calculus.

  2. General metric spaces, completeness, and uniform convergence.

  3. Two applications of the contraction mapping principle:

We will also cover some additional material on ODE's and multivariable calculus as time permits and interest warrants. This first part serves to provides the analytic foundations for the two theorems listed above, which are used in Differential Geometry.

The second part covers the Riemann integral and the Lebesgue integral, and the basic convergence theorems for the Lebesgue integral. We will also include the beginnings of general measure theory.

The analysis sequence at Stony Brook is unusual in that it defers the general topology to a separate Geometry/Topology sequence, and at the same time covers geometric subjects like ODE's and the inverse function theorem. Thus many of the standard textbooks, like Folland's book , Royden's Real Analysis or Rudin's Real and Complex Analysis are not quite appropriate; we use Geller because it closely follows the official syllabus.

Official curriculum sources:

The official math department MAT 544 Syllabus