Topology of metric spaces

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

This EU will address the following points:

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

- Connectivity: definition, connected sets of R. Continuous image of a connected set. Arc connectivity, convexity in a normed vector space. Connected components.

- Compactness: definition. Compact sets in^Rn. Continuous image of a compact set. Bolzano-Weierstrass theorem. Ascoli theorem.

- Completeness: Cauchy sequences in metric spaces, 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 set, dual.

The L1, L2, and first semester of L3 analysis courses, 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: -