ECTS
7 credits
Component
Faculty of Science
Description
This course develops the classical theory of Banach spaces and is also an introduction to spectral analysis.
Objectives
Master the basic tools common to all branches of analysis.
Necessary pre-requisites
A Bachelor's degree in Mathematics.
Recommended prerequisites: the content of the two L3 courses "Topology of metric spaces" and "Measure, integration, Fourier" of the Licence de Mathématiques of the University of Montpellier.
Syllabus
- Reminder of the Licence on Banach spaces and Hilbert spaces.
2. Baire spaces, Banach-Steinhaus theorem, open application and closed graph theorems.
3. Spaces C(K) : Ascoli theorem, Stone-Weierstrass theorem.
4. Hahn-Banach theorem: analytic form, geometric form.
5. Duality and weak topologies : topological dual, weak topology, weak topology*, notion of reflexive space.
6. Spectral analysis : compact operators, spectral decomposition.
Additional information
Hourly volumes:
CM : 24h
TD : 24h
TP : 0
Land : 0