Mathematical analysis : complements

linma1315  2018-2019  Louvain-la-Neuve

Mathematical analysis : complements
5 credits
30.0 h + 22.5 h
Q2
Teacher(s)
Absil Pierre-Antoine (compensates Van Schaftingen Jean); Absil Pierre-Antoine; Van Schaftingen Jean;
Language
French
Prerequisites
Basic calculus and linear algebra, such as taught, for example, in
LFSAB1101 (Mathématiques I) et LFSAB1102 (Mathématiques II)

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, partila differenial equations, optimal control, scientic 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 finitedimensional
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
MAP


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

Title of the programme
Sigle
Credits
Prerequisites
Aims
Bachelor in Engineering
FSA1BA
5
LEPL1101 AND LEPL1102 AND LEPL1105