MAT452 - Analyse fonctionnelle (2023-2024)

This course intends to provide the foundations of Functional Analysis having in mind applications to partial differential equations and applications to 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: study of convex sets, Helly's Theorem, Hahn-Banach's convex separation Theorem, Krein-Milman's Theorem.

Then, it continues with the study of theorems which form the basis of functional analysis: Baire's Lemma, Banach-Steinhaus Theorem, Open Mapping Theorem and the Closed Graph Theorem.

We then open an important chapter on the study of weak topologies and weak-∗ topologies, which will lead us to the statement of the Banach-Alaoglu Theorem (which allows us to “recover" some compactness in infinite dimensional spaces ).

After getting a little lost in the study of very general Banach spaces, we will see to what extent “reflexive” spaces and “separable” spaces constitute an interesting class of Banach spaces, since they enjoy pleasant properties.

The next chapter is devoted to the study of Banach algebras which unify under the same banner several special cases that you may have already seen (e.g. exponential of a matrix or a linear map). This chapter culminates with the proof in three lines (but which requires having understood the previous 10 pages!) of a beautiful result on Fourier series.

The course ends with the study of the spectrum of operators with in particular the Fredholm alternative, the study of the spectrum of compact operators to culminate with the study of Fredholm operators, which generalize in infinite dimension, the results that you know well of course linear maps between finite-dimensional vector spaces.