Duration
30h Th, 10h Pr, 20h Mon. WS
Number of credits
Lecturer
Language(s) of instruction
French language
Organisation and examination
Teaching in the first semester, review in January
Schedule
Units courses prerequisite and corequisite
Prerequisite or corequisite units are presented within each program
Learning unit contents
The course develops the fundamental principles of functional analysis and presents several of their applications.
It introduces topological vector spaces and locally convex spaces, in particular through families of seminorms, as well as continuous linear operators between such spaces. Several constructions are considered, including subspaces, product and quotient spaces, and some projective and inductive limits.
The course then addresses completeness, Banach and Fréchet spaces, together with several fundamental theorems of functional analysis: the Banach-Steinhaus theorem, the open mapping theorem, and the closed graph theorem.
Bounded, precompact, and compact sets are also studied, together with the Arzelà-Ascoli theorem. Finally, the course introduces the topological dual, the Hahn-Banach theorem, and weak topologies, through various classical examples of function spaces.
Learning outcomes of the learning unit
By the end of the course, the student will be able to understand and explain the fundamental concepts of functional analysis, present the main theorems, and apply functional analysis techniques to solve a variety of problems.
Prerequisite knowledge and skills
A good understanding of previous courses in analysis, linear algebra, and general topology is essential.
Planned learning activities and teaching methods
The course includes lectures, delivered either in person or online, as well as exercise sessions.
During the lectures, the main theoretical results are introduced, established, and illustrated with examples. Some parts of the course material may be studied independently, in particular using the available videos and course notes.
During the exercise sessions, students are trained to solve a variety of problems independently, using the results presented in the course. Some exercises may also be prepared or worked on independently.
The precise organisation of these activities may be adapted throughout the semester, depending on the progress of the course and the students' needs.
Mode of delivery (face to face, distance learning, hybrid learning)
Blended learning
Further information:
Combination of face-to-face, online, and independent learning activities. The precise arrangements will be adapted throughout the semester depending on the progress of the course and the number of students enrolled.
The course is offered in alternating years and will not be taught in 2027-2028. It will nevertheless remain possible to follow the course independently using the available videos and course notes; no other learning activities will be organised during that academic year.
Course materials and recommended or required readings
Lecture notes and a list of reference works are available on eCampus.
Exam(s) in session
Any session
- In-person
written exam ( open-ended questions ) AND oral exam
Further information:
The exam consists of two parts.
The written part focuses on solving problems related to the theoretical material covered in the course.
The oral part consists of a presentation at the blackboard on a topic from the course. Two lists of topics will be provided. The student must choose one topic from each list. At the time of the exam, the instructor will select which of these two topics the student will present.
Work placement(s)
Organisational remarks and main changes to the course
The course is offered during the first quadrimester every other academic year, in academic years beginning in an even-numbered year.
The course is offered in alternating years and will not be taught in 2027-2028. However, it can still be followed independently using the available videos and lecture notes; no other learning activities are planned for that year.
Contacts
Céline Esser
Email : Celine.Esser@uliege.be
Département de Mathématique,
Allée de la Découverte, 12, B37,
4000 Liège Belgium
Bureau 1/75
Association of one or more MOOCs
Items online
Course web page
Web page giving access to various informations on the course and to the electronic version of the notes.