Topology of metric spaces

Introduce the basic concepts of topology and their use in the study of functional spaces.

This UE will cover the following points:

- Metric and topological spaces: definition, limits and continuity. Open, closed, neighborhoods. Interior and adherence of a part, density. Product and quotient topology.

- Connectedness: definition, connexes of R. Continuous image of a connexe. Connectedness by arc, convexity in a normed vector space. Related components

- Compactness: definition. The compacts of ^Rn. Continuous image of a compact. Bolzano-Weierstrass theorem. Ascoli theorem.

- Completeness: Cauchy sequences in a metric space, definition of a complete metric space. Extension of applications, completion of a metric space. Fixed point theorem.

- Banach and Hilbert spaces: definition, the case of finite dimension. Continuous linear applications, topological dual. Examples: ^Lp and ^C0 spaces. Hilbert spaces, projection onto a closed convex, dual.

Analysis courses in L1, L2 and the first semester of L3, in particular :

- HAX404X Topology of ^Rn and functions of several variables

- HAX502X Differential calculus and differential equations

Recommended prerequisites: first semester of L3

Hourly volumes :

            CM: 31.5

            TD: 31.5

            TP: -

            Land: -