Mathematics 797FN - ST-Functional Analysis & Appl

The main goal of this class is to develop a number of tools from functional analysis and to show how they are used in various contexts by considering concrete examples (from partial differential equations, probability, dynamical systems, etc...). Among the topics covered in this class are: (i) Banach and Hilbert spaces, Linear functionals, Dual Spaces, Hahn-Banach Theorem. (ii) Linear operators (bounded and unbounded), open mapping and closed graph theorem, spectrum, Banach algebras, functional calculus. (iii) Spectral theory for compact operators, Fredholm theory, positive operators. (iv) Spectral theorem for unitary and self-adjoint operators. (v) Semigroups and unbounded operators. Prerequisites: Math 623-624 or equivalent. We assume a basic knowledge of Banach spaces and Hilbert spaces , (definition, examples, bounded linear operators, dual spaces, orthonormal basis, projection, etc.) as usually taught during Math 623/624. These concepts will be reviewed during the class but at a fast pace. We will use a little (just a little) measure theory. In any doubt contact the instructor.

Open to Graduate students only.