Extended description for Functional Analysis: Math 5130-5131: Topics in functional analysis, at the choice of the instructor and students, including the following.

Theory of Banach spaces: duality, reflexivity, weak and weak* topologies. Hahn-Banach, Banach-Steinhaus, Banach-Alaoglu theorems. Krein-Milman theorem. Linear bounded operators on Hilbert spaces: compact, integral, trace class, Fredholm, Hilbert-Schmidt, Toepliz, Volterra. Compact self-adjoint operators with applications to the classical Sturm-Liouville theory. Spectral theory of unbounded self-adjoint and normal operators on Hilbert spaces, quadratic forms. Examples and counterexamples of self-adjoint operators, and the role of the Fourier transform. Spectral theory of differential operators with constant coefficients. Unitary and positivity preserving operator semigroups, resolvents, Trotter product formula, Hille-Yosida theorem. Other topics may include distributions, commutative Banach and C* algebras, Gelfand transform and its relation to the spectral theorem for bounded normal operators, Banach algebra L¹, Kaplansky density theorem, Gelfand-Naimark-Segal construction, introduction to von Neumann algebras and non-commutative integration. Applications to probability, quantum mechanics, Kigami's resistance forms and Dirichlet forms.

Textbook choices: Functional Analysis by W. Rudin. A Course in Functional Analysis by J.B.Conway. Functional Analysis by P.D. Lax. Functional Analysis (Methods of Modern Mathematical Physics, Volume 1) by M. Reed and B. Simon. Spectral Theory of Self-Adjoint Operators in Hilbert Space by M.S.Birman and M.Z.Solomjak. Supplementary reading: Functional Analysis by K. Yosida. Functional Analysis, Sobolev Spaces and Partial Differential Equations by H. Brezis. Lecture notes by A.J. Wassermann and by V.F.R. Jones. Lectures on Quantum Mechanics for Mathematics Students by L.D. Faddeev and O.A. Yakubovskii.
Prerequisites: MATH 5111
Credits: 3