ECTS
7 credits
Component
Faculty of Science
Description
Introduce the basic notions of topology and their use for the study of functional spaces.
Objectives
This EU will address the following:
- Metric and topological spaces: definition, limits and continuity. Open, closed, neighborhoods. Interior and adherence of a part, density. Product topology and quotient topology.
- Connectedness: definition, connectedness of R. Continuous image of a connectedness. 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's 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 on a closed convex, dual.
Necessary pre-requisites
The analysis courses of 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
Additional information
Hourly volumes:
CM: 31.5
TD: 31.5
TP: -
Land: -