Real Analysis I


Professor: Jeremy Kahn
Office hours: Monday 11:20-12:40 and Thursday 12:50-2:10.

Textbook: Basic Real Analysis by Knapp.
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 assignements will usually be available at this website by Monday evening. Knapp has "hints to solutions" to all the problems in the book at the back, which usually give a complete outline to the solution. You are welcome to use these hints; excessive dependence on them will be exposed on the midterm, the final, and on the comps. 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 covergence 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 Royden's Real Analysis or Rudin's Real and Complex Analysis are not quite appropriate; we use Knapp because it closely follows the official syllabus (Knapp was a professor at Stony Brook).

Official curriculum sources:

The official math department MAT 544 Syllabus
The graduate bulletin (with offical SUNY Stony Brook syllabi).