# MAT 551: Real Analysis III

Fall 2011

Department of Mathematics

SUNY at Stony Brook

**Textbooks**:

- Michael Reed and Barry Simon,
*Method
of Modern Mathematical Physics*, v. 1 & 2, Academic Press, 1972

- Daryl Geller,
*A
first graduate course in real analysis. Parts I-II* Solutions Custom Publishing (can be
ordered from the campus bookstore)

You can also consult many other good textbooks and monographs on
functional
analysis, including:

- N. Akhiezer and I. Glazman, Theory of linear operators in Hilbert
space.
*Dover Publications, Inc., *New York, 1993

- N. Dunford and J. Schwartz, Linear operators. Part I.
General theory.
*John Wiley & Sons, Inc., *New
York, 1988

- K. Iosida, Functional analysis. 6th ed.,
*Springer-Verlag, *Berlin-New
York, 1980

- W. Rudin, Functional analysis, 2nd ed.,
*McGraw-Hill,
Inc., *New York, 1991

**Schedule: TuTh** 12:50 - 2:10 PHYSICS P125.

**Instructor:** Leon Takhtajan, Math Tower 5-116, Phone:
632-8290, email: *leontak@math.sunysb.edu.*

**Topics covered:**

- Review of Banach, Hilbert and locally convex vector spaces
- Distributions and Fourier transforms, Hermite functions
- Bounded and compact operators, the spectral theorem
- Unbounded operators, symmetric and self-adjoint operators,
general
spectral theorem, examples
- Stone theorem, the Trotter product formula
- Commutative Banach algebras, C*-algebras

**Prerequisites: **The basic core courses curriculum.
**Howework & Exams: **Homework will be posted on the web
page,
with problems varying from routine to more challenging. Course grades
will
be based on these problems, class participation, the midterm exam and
the
final exam.

- Review of Real Analysis I and II.
- HW 1 Ch. 2 in Reed-Simon, v.1:
4,5,16,22,25.
- Introduction to locally convex topological vector
spaces and theory of distributions.
- HW 2 Ch. 5 in Reed-Simon, v.1:
1,4,16,21,22,24,25,45,47.
- Notes on Fourier Transform.
- HW 3 Ch. 6 in Reed-Simon, v.1:
1,5,7,9,10,11,14,22.
- Notes on bounded and compact operators.
- Notes on trace class and Hilbert-Schmidt
operators.
**Take
Home Exam.
****Notes on spectral theorem. ****
**- Notes on unbounded operators.

**
**

*DSS advisory.* If you have a physical,
psychiatric,
medical, or learning disability that may affect your ability to carry
out
the assigned course work, please contact the office of Disabled Student
Services (DSS), Humanities Building, room 133, telephone 632-6748/TDGD.
DSS will review your concerns and determine what accommodations may be
necessary and appropriate. All information and documentation of
disability
is confidential.