Foundations of Advanced Analysis (mod. 1)

Foundations
of Advanced Analysis (mod. 1)

Credits

6

Prerequisites

None.

Examination method

Passing an integrated examination, possibly divided into several tests, on the contents of Foundations of Advanced Analysis mod. 1 and Foundations of Advanced Analysis mod. 2

Contents

Introduction to complex variables: holomorphic and analytic functions, Cauchy-Riemann equations, Cauchy’s theorems, residue theory. Introduction to measure theory: construction of the Lebesgue measure in Rⁿ and proof of its main properties (inner and outer regularity, invariance under rigid motions). Examples of non-measurable sets. Introduction to abstract measures and integration theory. Theorems on passing to the limit under the integral sign. Riesz representation theorem for positive linear functionals. Density of continuous functions. Jensen, Hölder and Minkowski inequalities. L^p spaces. Introduction to product measures and the Fubini and Tonelli theorems. Introduction to complex measures, Radon-Nikodym theorem. The duals of L^p. Maximal function and differentiation of measures. Change of variables. Lebesgue points, almost everywhere differentiability of monotone functions. Introduction to absolutely continuous functions and the fundamental theorem of integral calculus. Fourier transform in L¹ and L². Introduction to functional analysis: Hilbert spaces: projections onto a convex set, representation theorem for continuous linear functionals, orthonormal systems, Bessel’s inequality, Fourier series. Banach spaces: Hahn-Banach theorem, open mapping theorem, closed graph theorem.