Advanced Real Analysis
Math 212b / Tu Th 1011:30 / 216 SC
Harvard University  Spring 2006
Instructor:
Curtis T McMullen
(ctm@math.harvard.edu)
Required Texts
 Rudin,
Functional Analysis, 2nd ed.
McGrawHill, 1991
 Berberian,
Lectures on Functional Analysis.
SpringerVerlag, 1974 (unfortunately, out of print)
Recommended Texts
 RieszNagy,
Functional Analysis.
Dover reprint, 1990.
 Reed and Simon,
Functional Analysis I.
Academic Press, 1980.
Related Literature
 Stein,
Singular Integrals and Differentiability
Properties of Functions.
Princeton University Press, 1970.
 de la Harpe et Valette,
La Propriété (T) de Kazhdan pour les
Groupes Localement Compacts.
Astérisque 175, 1989;
distributed by the American Mathematical Society.
Prerequisites.
Directed to graduate students.
Lebesgue measure and integral, general topology,
and theory of linear operators between Banach spaces
will be assumed.
Topics.
This course will present several topics,
from elliptic partial differential equations
to ergodic theory,
with spectral theory and unitary representations as
an underlying theme. Possible topics include:
 Distributions and Fourier Analysis

Test functions, convolutions

Distributions and duality

Fourier transform and Sobolev spaces

Singular integral operators

Elliptic equations

Fluid flow in the plane

Prime number theorem
 Operator algebras

Banach algebras and C* algebras

The Gelfand representation

Functional calculus

Positive operators

Selfadjoint operators

Unitary operators and representations

Ergodic theory, Property T, expanding graphs,
and further topics
Grades.
Graduate students who have passed their
quals are excused from a grade for this course.
Grades for other students will be based on homework and exams.
Collaboration is encouraged on homework.
Exam work should be based only on course materials and done
without the help of others.
2006 Calendar.
2 Feb (Th) 
First class. 
28, 30 Mar (Tu,Th) 
No class  spring break. 
4 May (Th) 
Last class. 
Course home page: