SPOCs available at Sorbonne University - Basic functional analysis

Basic functional analysis - 4M105

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. A solid mastery of the contents of this chapter is absolutely imperative. 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. It may be short, but it is fundamental. Duality is the basis of the theory of distributions, a major breakthrough in analysis at the start of the XXth 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

_{Domain Mathematics}

_{Prerequisites}

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

Workload - 150 hours in total