**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 textbook***Real
Analysis: Modern Techniques and Their Applications (2nd edition) by
Gerald B. Folland *

**Final Exam: Tuesday, December 15, 2:15PM-4:45PM Mathematics 4--130**

Syllabus

**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:

Review of rigorous foundations of calculus.

General metric spaces, completeness, and uniform convergence.

Two applications of the contraction mapping principle:

The existence and uniqueness theorem for ordinary differential equations

The inverse function theorem

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.