Mathematical analysis : complements

linma1315  2019-2020  Louvain-la-Neuve

Mathematical analysis : complements
Note from June 29, 2020
Although we do not yet know how long the social distancing related to the Covid-19 pandemic will last, and regardless of the changes that had to be made in the evaluation of the June 2020 session in relation to what is provided for in this learning unit description, new learnig unit evaluation methods may still be adopted by the teachers; details of these methods have been - or will be - communicated to the students by the teachers, as soon as possible.
5 credits
30.0 h + 22.5 h
Q2
Language
French
Prerequisites

The prerequisite(s) for this Teaching Unit (Unité d’enseignement – UE) for the programmes/courses that offer this Teaching Unit are specified at the end of this sheet.
Main themes
This course covers themes in mathematical analysis (measure theory, functional analysis and function spaces) that play a role in the foundations of various areas of applied mathematics such as dynamical systems, partial differential equations, optimal control, scientific computing, stochastic processes and financial mathematics.
Aims

At the end of this learning unit, the student is able to :

1

AA 1.1, 1.2, 1.3, 3.1.
At the end of the course, the student will be able to:
1. by means of examples, statements and mathematical proofs, describe infinite-dimensional spaces, including their operators and convergence notions, and compare them to finite dimensional spaces,
2. apply definitions and results of measure theory to the study of function spaces and probability theory,
3. use advanced concepts of measure theory and functional analysis in applied mathematics.

 

The contribution of this Teaching Unit to the development and command of the skills and learning outcomes of the programme(s) can be accessed at the end of this sheet, in the section entitled “Programmes/courses offering this Teaching Unit”.
Content
Important concepts and results within the main themes of the course,
such as:
  • Measure theory, Lebesgue integral, convergence theorems,
  • Complete metric spaces, Banach spaces and Hilbert spaces, spaces of continuous functions, spaces of integrable functions,
  • Continuous linear mappings, weak convergence, Riesz representation theorem, notions of spectral theory,
  • Distributions and Sobolev spaces.
Teaching methods
The course includes interactive lectures and exercises. The emphasis is
on critical understanding of the theory and active problem solving.
Evaluation methods
  • Homeworks, exercises, tests or practical work carried out during the semester
  • Exam
More elaborate information on the evaluation procedure is given in the
course outline, made available on Moodle at the beginning of the
academic year.
Other information
Bibliography
Livre de référence : Gerald Teschl, "Topics in Real and Functional Analysis" disponible gratuitement en ligne à l'adresse
(https://www.mat.univie.ac.at/~gerald/ftp/book-fa/).
Faculty or entity


Programmes / formations proposant cette unité d'enseignement (UE)

Title of the programme
Sigle
Credits
Prerequisites
Aims
Minor in Applied Mathematics

Specialization track in Applied Mathematics