ECTS
7 credits
Training Structure
College of Sciences
Description
This course unit covers the classical theory of Banach spaces and also serves as an introduction to spectral analysis.
Objectives
Master the basic tools common to all branches of analysis.
Class Hours
- Functional Analysis - LectureLecture24 hours
- Functional Analysis - TutorialTutorials24 hours
Mandatory Prerequisites
A bachelor's degree program in mathematics.
Recommended prerequisites: the material covered in the two third-year courses, “Topology of Metric Spaces” and “Measure, Integration, and Fourier,” in the Bachelor’s program in Mathematics at the University of Montpellier.
Course Outline
- Review of Undergraduate Concepts on Banach Spaces and Hilbert Spaces.
2. Baire spaces, the Banach-Steinhaus theorem, the open mapping theorem, and the closed graph theorem.
3. C(K) spaces: the Ascoli theorem, the Stone-Weierstrass theorem.
4. The Hahn-Banach theorem: analytic form, geometric form.
5. Duality and weak topologies: topological dual, weak topology, weak topology*, concept of a reflexive space.
6. Spectral analysis: compact operators, spectral decomposition.
Additional Information
Hourly volumes:
CM: 24 hours
TD: 24 hours
TP: 0
Land: 0