ECTS
7 credits
Training structure
Faculty of Science
Description
This course develops the classical theory of Banach spaces and also provides an introduction to spectral analysis.
Objectives
Mastering the basic tools common to all branches of analysis.
Teaching hours
- Functional Analysis - CMLecture24 hours
- Functional Analysis - TutorialTutorials24 hours
Mandatory prerequisites
A Bachelor's degree in Mathematics.
Recommended prerequisites: the content of the two L3 courses "Topology of Metric Spaces" and "Measure, Integration, Fourier" from the Bachelor's degree in Mathematics at the University of Montpellier.
Syllabus
- License reminders on Banach spaces and Hilbert spaces.
2. Baire spaces, Banach-Steinhaus theorem, open mapping theorem and closed graph theorem.
3. C(K) spaces: Ascoli's theorem, Stone-Weierstrass theorem.
4. Hahn-Banach theorem: analytical 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: 24 hours
TD: 24 hours
TP: 0
Land: 0