The objective of the course is to study the fundamental concepts and the main theorems of functional analysis in the framework of infinite dimensional normed spaces and to apply them to the solution of linear functional equations (existence and uniqueness of solutions, convergence of iterative schemes, spectral decomposition).
After defining linear and bounded operators, the main classical results are reviewed: Hahn-Banach theorems, convex separation theorems, Banach-Steinhaus theorem, open application and closed graph theorems, study of weak topology and weak topology-*, Banach-Alaoglu theorem, reflexive spaces. The second part of the course is devoted to the spectral theory of compact operators and its application to integral equations. After having seen the Neumann lemma and the perturbation lemma, we examine in detail the Fredholm alternative and its interpretation in terms of integral equations. Then we particularise the results obtained to the case of Hilbert spaces to obtain the spectral decomposition theorem.
The examination consists of two parts: a written examination (with a mark of 35% of the overall mark) and an oral examination (with a mark of 40% of the overall mark). In addition, the work
The personal mark is 25% of the overall mark. Each of these three marks corresponds to a learning activity.
The questions in the written examination are exercise questions only: they are based on applications of the same kind as those proposed in the tutorials and in the course. They are designed to assess the student's ability to apply the main concepts and results of the course.
In the oral examination, the questions are on theory. The emphasis is on understanding, accuracy and synthesis. This examination consists of two main questions, one
The other is a knowledge question which consists of giving definitions and stating results (without demonstration).
on a specific theme.
The student has one hour to prepare for these two questions. The presentation of the questions, including the answers to the sub-questions, may never exceed thirty minutes.
Students will be expected to know all the concepts, definitions and statements of all the results (theorems, propositions, etc) presented in the course. Demonstrations will only be required for results from the list provided in writing.
Precise instructions will be given in due course.
Functional analysis. Theo Bühler and Dietmar A. Salamon. American Mathematical Society. 2018. Functional analysis. Theory and applications Haim Brezis Masson Paris 1983.
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer 2011.
| Training | Study programme | Block | Credits | Mandatory |
|---|---|---|---|---|
| Bachelor in Computer Science | Standard | 0 | 5 | |
| Bachelor in Computer Science | Standard | 2 | 5 |