Instructor: Dr. Shuzhou Wang

Functional analysis is a vastly developed subject of mathematics with broad applications in pure and applied mathematics, as well as in other disciplines such as physics, statistics, economics, engineering and operations research, to name a few. It also provides foundations for active research areas such as operator algebras, operator theory, noncommutative geometry and quantum groups. We will pace this course to develop understanding of basic aspects of the theory instead of covering large amount of material rapidly. We hope this course will lay the foundation for those interested in delving deeper into the subject, as well as for those who wish to use the subject as a tool in other fields. This course should be accessible to those with background in the theory of calculus and linear algebra, such as the theory of limits and continuity, linear transformations over vector spaces. For those without the background but interested in the subject, a willingness to take on faith a few facts without proofs will help. 

Topics of this course will consists of three parts.

Part 1: Banach spaces and examples, the open mapping theorem, the closed graph theorem, various versions of the Hahn-Banach theorem. Weak *-topologies, Alaoglu theorem, Krein-Milman theorem.

Part 2 : Hilbert spaces and examples, Riesz representation theorem, orthonormal bases, projection operators, unitary operators, self-adjoint operators, normal operators.

Part 3: Commutative Banach algebras and examples, characters and maximal ideal space, Gelfand-Fourier transform, function algebras, Stone-Weierstrass theorem, Gelfand-Naimark theorem for commutative C*-algebras, the spectral theorem for normal operators.

As a sequel to this course, I plan to teach a topics course (Math 8130) on functional analysis in Fall, 2006, which will serve as a bridge to current research. Topics in that course will be selected from C*-algebras, von Neumann algebras and quantum groups.