Real Analysis
Math 531  SM3:30PM4:45PM
King Fahd University of Petroleum and Minerals 072
Real Analysis is an enormous field with applications to many areas of mathematics. Roughly speaking, it has applications to any setting where one integrates functions, ranging from harmonic analysis on Euclidean space to partial differential equations on manifolds, from representation theory to number theory, from probability theory to integral geometry, from ergodic theory to quantum mechanics.
Instructor: Dr. Assane Lo
Required Text
 Royden,
Real Analysis, 3rd ed.
PrenticeHall, 1988.
Recommended Texts
Prerequisites.
Intended for graduate students.
Topics.
This course will provide a rigorous introduction to
measurable functions, Lebesgue integration, Banach spaces
and duality.
Possible topics include:

Functions of a real variable

Real numbers; open sets; Borel sets; transfinite induction.

Measurable functions. Littlewood's 3 principles.

Lebesgue integration

Monotonicity, bounded variation, absolute continuity.

Differentiable and convex functions.

The classical Banach spaces.

Classical Banach spaces

The Lp spaces.

The Minkowski and Holder inequalities.

Convergence and Completeness.

Approximation in Lp.

Bounded linear functionals on the Lp spaces.
 General Measure and Integration Theory
 Measure spaces.
 Measurable Functions.
 Integration.
 General convergence Theorems.
 Signed measures.
 The radonNikodym Theorem
 The general Lp spaces
Reading and Lectures.
Students are responsible for all topics covered in
the readings and lectures.
Assigned material should be read before
coming to class. Lectures may go beyond the
reading, and not every topic in the reading will be
covered in class.
Grades.
The grade will be based on HW (50%), the midterm (25%) and the final (25%) .
Homework. Homework will be assigned once every
week or two. Late homework will not be accepted. Collaboration between students is encouraged, but you
must write your own solutions, understand them and
give credit to your collaborators.
Exams. There will be one midterm and a final.
