Functional Analysis

Credits

Prerequisites

None.

Scientific-disciplinary sector (SSD)

MAT/05 Mathematical Analysis.

Examination method

Oral exam.

Learning
objectives

The main objectives of the course are the ability to set up the study of mathematical-physical models in a framework of function spaces, to analyse the most relevant properties of these spaces, and to carry out this study with the tools acquired.

Syllabus

Outline of mathematical-physical models. Review of vector and topological spaces. Compactness. Normed spaces and their completion. Compactness criteria in function spaces. Compactness and characterisation of finite-dimensional spaces. Fixed-point problems and the Leray-Schauder theorem. Applications to nonlinear ordinary differential equations. Minkowski functionals. Extension of linear functionals. Separation of convex sets. Topological vector spaces. Identity between locally convex spaces and spaces with a topology generated by families of seminorms. Banach spaces. Weak and weak-star topologies and compactness criteria. Minimisation of convex functionals. Applications to problems in the calculus of variations. Spaces of linear operators. Principles of uniform boundedness, the open mapping and the closed graph. Closed-range operators. Semi-Fredholm operators. Compact operators. Riesz-type operators. Adjoint of an operator. Complementarity relations. Fredholm operators. Index. Spectrum. Index and spectrum of a Riesz-type operator. Hilbert spaces and orthogonal projections. Representation of linear functionals. Diagonalisability of compact self-adjoint operators. Applications to the wave equation and the heat equation.

Expected learning
outcomes

By the end of the course, students must demonstrate that they

know and understand the specific language of the discipline as well as the general issues relating to the course topics;
are able to apply the knowledge acquired to the study and solution of concrete examples, making correct use of proof techniques;
are able to communicate ideas and solutions clearly, rigorously and effectively to both specialist and non-specialist audiences;
are able to identify the most appropriate methods to analyse and solve a problem relating to the course topics and to 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.

Functional Analysis