FMA_3X061_EP - Introduction à l analyse réelle (2024-2025)

The FMA_3X061_EP lecture focuses on the basics of real and functional analysis. There will be four parts:

1. Topology of normed vector spaces
2. Differential equations
3. Integration
4. Hilbert spaces and applications.

All these topics have a strong connection with other courses (physics, applied mathematics, mecanics for example). 

The content will be as follows: we shall start with topology of metric spaces and especially normed vector spaces. The main definitions will be those of connected, compact and complete spaces with a strong connection with sequences and series. We will especially consider the case of spaces of functions where we will state some of the fundamental theorems in functional analysis: approximation of functions by simpler ones (e.g. polynomials) and fixed points theorems.

We will use Fixed Points Theorems to prove the local existence of solutions of differential equations (Cauchy-Lipschitz Theorem) and extension of solutions. After having studied some explicit methods to find solutions, we shall explain how to get qualitative informations on solutions without having an explicit solution.

Intégration theory is also considered in the Applied Mathematics lecture. We will only review the main definitions and results of Lebesgue integration. This will be used to define new spaces of functions for which the first part of the lecture can be applied and are usefull in many situations. 

The last part will deal with the study of Hilbert spaces. Fourier analysis (Fourier series and Fourier transform) will be the main topic of this part.