MTH-109.4 Advanced Real Analysis
Construction of Lebesgue measure on \(\mathbb{R}^n\). \(\sigma\)− algebras, abstract measures. Quick overview of convergence theorems. Product measures, Fubini's theorem. Lebesgue decomposition, Radon-Nikodym theorem, \(L^p\) 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.