Mathematics

The course aims to provide advanced knowledge of both theoretical and practical programming in C++11 and Python3, with particular regard to the principles of object-oriented programming and best practices of software development.

Due to covid-19 containement rules, this year, videos with pre-recorded lectures will be uploaded weekly, starting from the week of October 5, 2020.

Aim of the course is to introduce the basic tools of linear and nonlinear functional analysis, and to show applications of such techniques in analysis. The course is divided into two parts: the first one concerns spectral theory of linear operators, whose goal is to extend the classical notion of spectrum of a matrix to an infinite dimensional setting. The second part of the course is an introduction to the methods of nonlinear analysis to find zeros of a nonlinear functional over an infinite dimensional space.

0. Metric and normed spaces, converging and Cauchy sequences, completeness, Banach spaces. Completion theorem. Dimension, finite-dimensional spaces and infinite-dimensional spaces. Space of real polynomials on an interval; space of real continuous functions on an interval. Compactness, sequential compactness, precompactness, relative compactness and their relations. Density and separability. Linear bounded operators on Banach spaces and their basic properties. Extension theorem.

The course aims to provide advanced knowledge of both theoretical and practical programming in C++11 and Python3, with particular regard to the principles of object-oriented programming and best practices of software development.

Syllabus: