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.

- Banach spaces: classical examples, completeness theorems, fundamental inequalities. Linear operators bounded between two spaces, linear forms, dual space.
- Hahn-Banach's theorem: axiom of choice, Zorn's lemma, and analytic form of the theorem. Bidual of a Banach space, reflexivity. Dual spaces ℓ
^{p}(N) and L^{p}(Ω). - 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.
- Theorems of the open application and of the closed graph. Topological supplement of a closed subspace, continuous projectors, inversible operators left or right.
- 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.
- 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 L
^{p}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

**Additional documentation**

- Daniel Li, Hervé Queffélec,
*Introduction à l'étude des espaces de Banach. Analyse et probabilités*, Cours Spécialisés 12, SMF, Paris, 2004 - Joram Lindenstrauss, Lior Tzafriri,
*Classical Banach spaces*(2 volumes), Springer, Berlin, 1979 - Michael Reed, Barry Simon,
*Methods of modern mathematical physics.**I. Functional analysis*, Academic Press, New-York, 1980 - Walter Rudin,
*Functional analysis*, McGraw-Hill, New York, 1991 (traduction française, Ediscience, Paris, 1995)