SPOCs available at Sorbonne University - 4M005

The bases of functional analysis - 4M005

Jean-Yves Chemin

_{Course description}

The course covers basic functional analysis in its own right together with an orientation towards applications to partial differential equations.

_{Full course description}

In the first chapter, we show basic results on the topology of metric spaces, including the notions of complete metric spaces and compact metric spaces.

The second chapter deals with the study of normed vector spaces, fundamental examples of which are function spaces. A key point here is understanding the effects of working in infinite dimension (which is the case in function spaces) on topology. Ascoli’s theorem, a compactness criterion for parts of continuous function spaces, illustrates the difficulties which appear in infinite dimension frameworks.

The third chapter deals with the notion of duality. Duality is the basis of the theory of distributions, a major breakthrough in analysis at the start of the XX^th century - this will be studied in chapter 8. Beyond the concept of transposes of linear maps, this chapter explains the procedure which allows one to identify the dual of a Banach space - another Banach space with a weaker notion of convergence, induced by the fact that it is a dual space, that we call “weak-star convergence”.

The fourth chapter is a classic: Hilbert spaces, which extend the notion of Euclidean spaces to infinite dimension.

The fifth chapter studies the spaces of functions which have a finite integral relative to a measure when elevated to a power p. We start by recalling fundamental results of integration theory, without proof. Another important notion, the convolution of functions, is defined, studied, then applied to approximation.

The sixth chapter studies the Dirichlet problem on a bounded domain.

The seventh chapter deals with the Fourier transform on the space of integrable functions on Rd and many applications. This chapter is fundamental and will be crucial in the following two.

The eighth chapter presents the theory of tempered distributions. We choose not to go into the general theory of distributions to keep things simple. The basic idea is that, when we know how to define an operation on very smooth and rapidly decreasing functions on Rd (e.g. functions in the Schwartz space), we can use duality to extend it to a space of tempered distributions, which contain functions (hence “distributions” are also called “generalised functions”) as well as some very singular objects. Of course, this chapter contains examples which must be studied and understood in order to properly comprehend and apply this theory.

_Domain

Mathematics

_{Prerequisites}

Linear algebra and topology of the third year of Bachelor's degree are imperative.

Workload - 300 hours in total