Math 610: Functional Analysis
My name is Maxim Gilula and I am the grader for this course. My office is DRL 3C13 and my office hours are by appointment. The instructor for this course is Dr. Charles Epstein.
The main goal of analysis is the solution of equations. In the first semester we covered 1-variable complex analysis; for most of this semester we turn to the analysis of linear equations in infinite dimensional. Unlike the finite dimensional case, this requires the introduction of a topology, and so becomes a topic in analysis. Amongst other things, we will develop the framework needed to find solutions of ordinary and partial differential equations.
We use Peter D. Lax, Functional Analysis as our principal text book. Good additional references for functional analysis are: Walter Rudin: Real and Complex Analysis and Functional Analysis; E. Stein and R. Shakarchi, Functional Analysis; Ward Cheney, Analysis for Applied Mathematics; Michael Reed and Barry Simon, Methods of Modern Mathematical Physics, Vol. 1; Tosio Kato, Perturbation Theory of Linear Operators. Good references for finite dimensional linear analysis are: Peter Lax, Linear Algebra.
A problem set will be assigned every week on Tuesday, due the following week on Tuesday. I very much prefer that students do the problem sets alone. We may have a take-home midterm and/or final exams.
Course Syllabus
(Borrowed from Dr. Epstein's webpage)
- Review of finite dimensional linear algebra
- Normed linear spaces, definitions and examples (1, 2, 5)
- Convexity, the Hahn-Banach Theorem (3, 4, 13.1)
- Hilbert space and the Riesz Representation Theorem (6, 7)
- Duality (8, 9)
- Weak convergence (10, 11)
- Bounded linear maps (15, 16, 20)
- Compact operators (21, 22)
- Fredholm theory (21)
- Spectra of compact operators in a Hilbert space (28, 29)
- Solving elliptic equations using boundary layer theory
Info and Homework Sets
Problem Set 1
Problem Set 2
Problem Set 3
Problem Set 4
Problem Set 5
Problem Set 6
Problem Set 7
Problem Set 8
Problem Set 9
