Prof. Orlando Merino, *merino@math.uri.edu*,
874 4442, Lippitt Hall 101C

MEETS: Tuesday and Thursday, from 3:30 to 4:45 pm, in room 204, Lippitt Hall

** MTH629
Functional Analysis** is the first of two one semester
courses on Functional Analysis. The course is
designed for students in
mathematics, science, engineering and other fields.
A firm foundation in Operator Theory is an essential part of mathematical training
for students of mathematics, as well as many other areas of
basic sciences and engineering.
This is an excellent course for
for graduate students
who would
like to strengthen mathematical skills and knowledge.

The main topic of MTH629 is the theory and applications of linear operators on a Hilbert space. That is, the course is a natural continuation of Linear Algebra. The applications include approximation and minimum norm problems, solutions to infinite systems of equations, and applications to integral and differential equations.

The requirements of the course are familiarity with linear algebra, advanced calculus, and mathematical proofs. Evaluation is based on homework and a take-home final exam.

The text is
**Basic
Classes of Linear Operators, by Gohberg, Goldberg, and Kaashoek**,
Birkhauser Verlag, 2003,ISBN: 978-3-7643-6930-9. The presentation of
material is elementary. It begins with finite dimensional vector spaces
and matrices, and the theory is gradually developed.

TOPICS of MTH 629 We will cover most of the following:

**Vector
Spaces**. Review of basic concepts.
Basis, subspaces, dimension, norm. Inner Product Spaces. Least squares
fit. Distance to convex sets. Orthonormal systems. Detour to Weierstrass
approximation theorem. Fourier Series.
**Bounded
Linear Operators on a Hilbert Space.**
Matrix representation. Bounded linear functionals. Operators of

finite rank.
Adjoint operators. Compact operators. Invertible
operators. Inversion by iterative method. Infinite systems of equations.
Application to integral equations.
**Compact
Self Adjoint Operators**. Existence
of eigenvalues and eigenvectors. The spectral theorem. Minimum-maximum
properties of eigenvalues. Integral Operators. Spectral
Theory. Sturm-Liouville systems.
**Applications**
to infinite systems of differential equations. Iterative Solutions
of Linear Equations and applications to integral equations.
**
Introduction to Banach Spaces and Nonlinear Operators**
Hahn-Banach theorem, Fixed point Theorems.