Advanced Analysis 2: Lebesgue Integration and Hilbert Spaces

This course is intended both for mathematics students continuing to honours work and for other students using mathematics at a high level in theoretical physics, engineering and information technology, and mathematical economics. Topics to be covered will normally be chosen from the following, with some additions and variations each year: - Measure and Integration - Lebesgue outer measure, measurable sets and integration, Lebesgue integral and basic properties, convergence theorems, connection with Riemann integration, Fubini's theorem, approximation theorems for measurable sets, Lusin's theorem, Egorov's theorem, Lp spaces, general measure theory, Radon-Nikodym theorem - Hilbert Spaces - elementary properties such as Cauchy Schwartz inequality and polarization, nearest point, orthogonal complements, linear operators, Riesz duality, adjoint operator, basic properties or unitary, self adjoint and normal operators, review and discussion of these operators in the complex and real setting, applications to L2 spaces and integral operators, projection operators, orthonormal sets, Bessel's inequality, Fourier expansion, Parseval's equality, applications to Fourier series. This is an Honours Pathway Course. It emphasises mathematical rigour and proof and develops modern analysis from an abstract viewpoint.