Credits: 9

Type: advanced

Objectives:

Introduction to functional analysis in the context of Banach spaces and their operators. Basic functional spaces, such as Lp-spaces and spaces of continuous function, with their duals, integral convolution operators, Sobolev distributions and spaces with applications to simple problems of differential equations.

Knowledge:

• Banach spaces and linear operators : Banach spaces and examples. Continuity of linear operators. Banach theorems. Examples.
• Duality: the Hahn-Banach theorem. The case of Hilbert spaces and the Radon-Nikodym theorem. The study of Lp-spaces and their duals.
• Distributions: Weak derivatives, Sobolev spaces. Applications to some elliptical problems.
• Spectral theory : Banach algebras. Spectral theory for selfadjoint operators.

Subject-specific competences:

To show understanding of abstract methods of functional analysis in specific contexts, such as those provided by integral operators and the results of regularization by convolution, and of the importance of distributions as tools for the resolution of equations.