Ce cours pose les fondements de l’analyse fonctionnelle, à la fois en amont des applications aux équations aux dérivées partielles, both in preparation for its applications to partial differential equations (elliptic, parabolic, or hyperbolic), and as a gateway to the study of operator algebras.
The objective is to provide a broad overview of the theory of Banach spaces and of operators defined between them.
The course begins with geometric considerations: the study of convex sets, Helly’s theorem, the Hahn–Banach separation theorem, and the Krein–Milman theorem.
It then continues with the core theorems that form the backbone of functional analysis: the Baire category lemma, the Banach–Steinhaus theorem, the open mapping theorem, and the closed graph theorem.
We next devote a key chapter to weak and weak* topologies, which naturally lead to the Banach–Alaoglu theorem — a fundamental result that recovers a form of compactness in infinite-dimensional spaces.
After this journey through the “jungle” of infinite-dimensional Banach spaces, we will focus on two particularly important classes: reflexive and separable spaces, which possess especially nice and reassuring structural properties.
The following chapter introduces Banach algebras, which provide a unified framework encompassing various classical examples (such as the exponential of a matrix or of a linear operator). This chapter culminates in an elegant three-line proof — resting on ten pages of theory! — of a beautiful result due to Wiener concerning Fourier series.
We will then study the spectrum of operators, beginning with the Fredholm alternative, followed by the spectral properties of compact operators, and concluding with the theory of Fredholm operators, which generalize classical results from linear algebra to the infinite-dimensional setting. In particular, we will define the Fredholm index, which extends the familiar identity dim Ker u + dim Im u = dim E for any linear map u: E --> F between finite-dimensional vector spaces.
Finally, the last chapter offers an introduction to unbounded operators and the theory of operator semigroups, which provide the natural framework for studying many evolution-type partial differential equations. In this context, we will prove the Hille–Yosida theorem (more precisely, one of its formulations known as the Lumer–Phillips theorem), which gives sufficient conditions for an operator to be the infinitesimal generator of a strongly continuous semigroup.
