MTH 629 Functional Analysis     Spring 2010      URI Department of Mathematics

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.