## UE Functional Analysis

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Degrees incorporating this pedagocial element :

### Description

The aim of this course is to demonstrate the main theorems that are the basis of analysis in Banach spaces of infinite dimension and to present a certain number of applications.

1. Banach spaces: classical examples, completeness theorems, fundamental inequalities. Linear operators bounded between two spaces, linear forms, dual space.
2. Hahn-Banach's theorem: axiom of choice, Zorn's lemma, and analytic form of the theorem. Bidual of a Banach space, reflexivity. Dual spaces ℓp(N) and Lp(Ω).
3. Baire's lemma and Banach-Steinhaus theorem. Low convergence of a sequence in a Banach space, and weak-star convergence of a sequence in its dual. Low sequential compactness of the unit ball of a reflexive space.
4. Theorems of the open application and of the closed graph. Topological supplement of a closed subspace, continuous projectors, inversible operators left or right.
5. Introduction to the spectral theory of bounded linear operators in a Banach space. Spectre, solving set, resolving operator, spectral ray. Possibly: spectral theory of compact operators.
6. Sobolev spaces in dimension one. Spaces H1(I) and H10(I) where I is a bounded interval: Poincaré inequality, injection in continuous functions, characterization using Fourier series. Space H1(R), characterization using the Fourier transform. Application to solving elliptic problems in dimension one.

### Prerequisites

• The course is based mainly on the topology of metric spaces and normed vector spaces studied in the first semester of L3. The language of general topology is also used.
• The integration theory seen in the second half of L3 is necessary to deal with classical examples such as Lp spaces.
• The Fourier series and the Fourier transform, which are included in the curriculum of the course Ordinary Differential Equations in the first half of the first year Master's, intervene in the study of Sobolev spaces.

### Bibliography

Main documentation

The contents of the course are entirely covered by H. Brézis's excellent classic work (in the original French version or in the English version, revised and expanded) below.

• Haïm Brézis, Analyse fonctionnelle, théorie et applications, Masson, Paris, 1983
• Haïm Brézis, Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, 2011