Prof. Orlando Merino     merino@math.uri.edu         874 4442

 
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.
    The main topic of MTH629 (Fall 98) is 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 and advanced calculus. Also, familiarity with mathematical proofs.

The text is the excellent Basic Operator Theory, by Gohberg and Goldberg. The presentation of material is elementary. It begins with finite dimensional vector spaces and matrices, and the theory is gradually developed.

The following course MTH630 (Spring 99) will be mainly on Banach spaces, inear and nonlinear operators on these spaces, and applications to optimization.

TOPICS
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. 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.
 

MTH 630 Spring 1999. (PRELIMINARY)

Normed and Banach Spaces.  Finite dimensional normed vector spaces. Separable spaces.  Dual spaces. The family of Hahn-Banach theorems.
Linear Operators on a Banach Space. Closed operators. Closed graph theorem. Spectrum of an operator.
Volterra integral operator. Complemented subspaces and projections. Duality.
Linear functionals on a Banach space. Dual space. Weak convergence.

Optimization in Hilbert Space.  Minimum norm problems. Normal equations. Dual problem. Pseudoinverse operators.
Optimization of functionals on normed spaces.  Gateaux and Frechet differentials. Extrema. Constrained optimization.
Concave and convex functionals. Non Linear Operators.

Fixed point theorems. Contraction mapping theorem.
Applications to the solution of integral and differential equations.

More optimization.  Newtons method for solving operator equations.  Semidefinite Programming.