FMA_43052_EP - Analyse fonctionnelle (2024-2025)

This course aims to provide the fundamentals of functional analysis, both as a precursor to applications in (elliptic, parabolic or hyperbolic) partial differential equations and as a precursor to applications in operator algebras.

The objective of the course is to give a fairly general overview of the study of Banach spaces and operators between Banach spaces.

The course begins with geometric considerations: studying convex sets, Helly's Theorem, Hahn-Banach Separation Theorem, and Krein-Milman Theorem.

It then continues with the study of the theorems that form the foundation of functional analysis: Baire's Lemma, Banach-Steinhaus Theorem, Open Mapping Theorem, and Closed Graph Theorem.

Next, we open an important chapter on the study of weak topologies and weak*-topologies, leading us to the statement of the Banach-Alaoglu Theorem (which allows us to "regain" some compactness in infinite-dimensional spaces).

After delving into the study of Banach spaces, we will see to what extent "reflexive" spaces and "separable" spaces constitute an interesting class of Banach spaces that enjoy pleasant properties.

The next chapter is devoted to the study of Banach algebras, which unify several particular cases you may have already encountered (for example, the exponential of a matrix or an endomorphism). This chapter culminates with the proof, in three lines (but requiring understanding of the previous 10 pages!), of a beautiful result concerning Fourier series.

The course continues with the study of the spectrum of operators, particularly Fredholm's alternative, the spectrum of compact operators, and concludes with the study of Fredholm operators, which generalize, in infinite dimension, results that you are familiar with concerning linear mappings between finite-dimensional vector spaces.

Finally, the last chapter of the course is dedicated to unbounded operators and the analysis of operator semigroups, which are the starting point for the study of many evolution equations. In this context, we will prove the Hille-Yosida Theorem (or rather one of its versions known as the Lumer-Phillips Theorem), which provides sufficient conditions for an operator to be the infinitesimal generator of an operator semigroup.