Fundamentals of Higher Analysis

Credits

Prerequisites

None.

Scientific-disciplinary sector (SSD)

MAT/05 Mathematical Analysis.

Examination method

Written exam (exercises and numerical problems, possibly multiple-choice) and oral exam.

Learning
objectives

The course presents some fundamental topics of Mathematical Analysis: an introduction to the theory of analytic functions, Lebesgue integration theory, Fourier series and the Fourier transform, and the first elements of Functional Analysis.

Syllabus

Introduction to the theory of functions of a complex variable, holomorphic functions and the Cauchy-Riemann condition, harmonic functions, power series and analytic functions, Cauchy’s theorem and formulae, infinite differentiability and analyticity of holomorphic functions, Taylor series expansion, zeros and identity principles, Laurent series expansion and study of isolated singularities, mean value property and maximum modulus principle, residue theory. Measure theory and abstract integration, passage to the limit under the integral sign; positive Borel measures on locally compact topological spaces, Riesz representation theorem, Lusin’s theorem; construction of Lebesgue measure on Rⁿ and its main properties, non-measurable sets. Jensen, Young, Hölder and Minkowski inequalities. L^p spaces, density of the class of simple functions and of the class of compactly supported functions. Notions of convergence for sequences of measurable functions. Measures on product spaces and the theorems of Tonelli and Fubini; convolutions. Introduction to complex measures, total variation measure, Radon-Nikodym theorem and Lebesgue decomposition, dual of L^p. Fourier transform in L¹ and L². Introduction to Functional Analysis: metric spaces, normed spaces, inner product spaces; linear operators and functionals. Hilbert spaces: projection onto a closed convex set and onto a closed subspace, representation of continuous linear functionals, orthonormal systems, Bessel’s inequality, Fourier series. The Hahn-Banach theorem and first consequences: dual of a normed space, bidual, weak convergence, reflexive spaces; separation of convex sets. Banach spaces: Baire’s theorem, Banach-Steinhaus theorem, uniform boundedness principle, open mapping theorem, closed graph theorem.

Expected learning
outcomes

On completion of the course, students must demonstrate that they

know and understand the issues relating to some fundamental chapters of mathematical analysis (complex variables, abstract integration and measure theory, L^p spaces, Banach and Hilbert spaces);
can apply the knowledge acquired to the study and solution of concrete examples, correctly using proof techniques;
can communicate ideas and solutions clearly, rigorously and effectively to both specialist and non-specialist audiences;
can identify the most appropriate methods to analyse and solve a problem relating to the course topics and interpret the results correctly.

Learning outcomes
to be assessed

Command of the knowledge acquired, clarity of presentation, rigour in the use of language, confidence in using the notions acquired.

Fundamentals of Higher Analysis