• ECTS

    7 credits

  • Component

    Faculty of Science

Description

This course develops the classical theory of Banach spaces and provides an introduction to spectral analysis.

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Objectives

Master basic tools common to all branches of Analysis.

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Necessary prerequisites

A Bachelor's degree in Mathematics.

 

 

Recommended prerequisites: the content of the two L3 courses "Topology of metric spaces" and "Measurement, integration, Fourier" from the Licence de Mathématiques at the Université de Montpellier.

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Syllabus

  1. Licence reminders on Banach spaces and Hilbert spaces.


2. Baire spaces, Banach-Steinhaus theorem, open application and closed graph theorems.

3. C(K) spaces: 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.

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Further information

Hourly volumes :

            CM : 24h

            TD : 24h

            TP: 0

            Land: 0

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