This series of lectures is an introduction to real and complex analysis.
The lectures are organized around 4 main topics :

- Lebesgue's theory of measure and integration ;
- Fourier analysis ;
- Hilbert spaces and variational methods ;
- the theory of holomorphic functions (functions of one complex variable which are complex differentiable).

This course is aimed at giving all students some basic knowledge in functional analysis, opening up the way to many different fields : pure mathematics, applied mathematics, mechanics , theoretical physics, ... In particular, this course is an important toolbox for the second and third year courses in pure and applied mathematics especially MAT431 and MAT432, which are the natural extensions of this course.

The theory of measure and integration developed by H. Lebesgue is a key tool in many branches of mathematics and is commonly used in applications (e.g. in numerical analysis). This theory also offers a natural framework to probability theory which is presented in the second year's course in applied mathematics (MAP 432) and is the foundation of geometric measure theory. Applications to Fourier analysis and in the contest of Hilbert spaces will be given. Fourier analysis finds many applications in the solvability of partial differential equations also in signal processing and (see course MAP 555),… Hilbert spaces theory is at the crossroad of analysis and geometry, it constitutes a first step towards the theory of operators and spectral theory. It is also an essential tool in the resolution of variational problems (see the course in optimization MAP 431) and partial differential equations (see the courses MAT431, MAT432 or MAP 431) used in physics as well as in mechanics (heat equation, wave equation Schoedinger's equation). The theory of holomorphic functions finds a variety of applications either in pure mathematics (number theory, minimal surfaces and geometry, ...) or in applied fields (fluid mechanics, …).

The mathematical concepts introduced in this course will be illustrated by applications : the use of Fourier analysis in the modeling of diffraction in optics, the use of Hilbert spaces in quantum mechanics and in variational problems, the use of holomorphic functions in fluid mechanics (aerodynamics) or in the study of minimal surfaces , ...

No particular prerequisite other than the background of "classes préparatoires" are needed to follow this course. The first lecture will discuss some complements in topology (topology of normed vector spaces and metric spaces) that are essential in Lebesgue's theory of integration and that Hilbert spaces.

Last Modification : Friday 21 October 2011