Real Analysis

Math 531 -- SM--3:30PM-4:45PM
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. Prentice-Hall, 1988.
Recommended Texts
  • Lieb and Loss, Analysis, Graduate Studies in Mathematics 14, AMS

  • Rudin , Real and Complex Analysis, McGraw Hill
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 radon-Nikodym 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 mid-term (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 mid-term and a final.