| Main Texts: | Horn and Johnson, Matrix Analysis
Coddington and Levinson, Theory of Ordinary Differential Equations |
Topics to be covered: Linear algebra (6-7 weeks): Review of undergraduate material from an abstract perspective, infinite dimensional vector spaces, norms, completeness, linear transformations and matrices, bilinear forms, finite dimensional spectral theory, normal forms, factorization theorems, resolvents, applications to optimization, least squares problems, numerical issues. ODE (4-5 weeks): existence and uniqueness results, linear systems, numerical approximations.
| Main Texts: | Coddington and Levinson, Theory of Ordinary Differential
Equations
Jones, Lebesgue Integration on Euclidean Space Riesz & Sz.-Nagy, Functional Analysis |
Topics to be covered: ODE (continued) -- linear systems, numerical approximations; overview of Lebesgue integration in Rn; Hilbert spaces; Fourier series with application to PDE's; Fourier transforms and convolutions.
| Main Texts: | Riesz & Sz.-Nagy, Functional Analysis
Friedlander, Introduction to the Theory of Distributions |
Topics to be covered: Fourier transforms and convolutions (continued);
time dependent PDE's and their numerical approximations; compact operators
with application to integral equations; spectral theory of bounded linear
operators with application to Sturm-Liouville problems; distribution theory.