Advanced Functional Analysis, Spectral Theory, and Applications

This course is intended both for mathematics students and for those using mathematics at a high level in theoretical physics, engineering and information technology, and mathematical economics. Selected topics from below will be covered: Hilbert spaces - orthogonality, bounded linear operators, Riesz representation, adjoints, unitary operators, compact operators, the spectral theorem for compact self-adjoint operators with applications. Measure theory - abstract measure theory, integration, signed measures, total variation, absolute continuity, the Lebesgue-Radon-Nikodym theorem, Hausdorff measure. Banach spaces - bounded linear operators, dual spaces, adjoints, the Hahn-Banach theorem, Baire category theorem and its consequences (uniform boundedness principle, closed graph and open mapping theorems), weak convergence, sequential version of Banach-Alaoglu theorem, interpolation. Applications to harmonic analysis and partial differential equations, such as the Fourier transform, maximal functions, singular integrals, Strichartz estimates. Note: Graduate students attend joint classes with undergraduates enrolled in MATH3325 but will be assessed separately.