Teacher(s)

Language

French

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.

Learning outcomes

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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. |

Content

Important concepts and results within the main themes of the course,

such as:

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.

on critical understanding of the theory and active problem solving.

Evaluation methods

- Work carried out during the term: homework assignments, exercises, or laboratory work. These activities are thus organized (and evaluated) only once per academic year.
- Exam: written, or sometimes oral depending on the circumstances.

Other information

Online resources

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/).

(https://www.mat.univie.ac.at/~gerald/ftp/book-fa/).

Faculty or entity