MTH-109.4 Advanced Real Analysis

Construction of Lebesgue measure on Rn. σ− algebras, abstract measures. Quick overview of convergence theorems. Product measures, Fubini's theorem. Lebesgue decomposition, Radon-Nikodym theorem, Lp

spaces, duality, Lebesgue differentiation theorem, absolutely continuous functions, monotone functions, Convolution and Fourier transform.

Normed Linear Spaces, Banach spaces, continuous linear functional, dual spaces. Quick overview of Hahn-Banach theorems, open mapping, closed graph and uniform boundedness theorems. Weak, weak∗ topologies, Banach-Alaoglu Theorem, reflexivity. Hilbert spaces, Riesz representation theorem, Spectral theorem for Compact operators.