NMAK10008U Functional Analysis (FunkAn)
MSc Programme in Mathematics
This course will cover a number of fundamental topics within the area of Functional Analysis. These topics include:
- Banach spaces: The Hahn-Banach theorem, including its versions as separation theorem, weak and weak* toplogies, the Banach-Alaoglu theorem, fundamental results connected to the Baire Category theory (the open mapping theorem, the closed graph theorem and the Uniform Boundedness Principle), as well as convexity topics, including the Krein-Milman theorem and the Markov-Kakutani fixed point theorem.
- Operators on Hilbert spaces, Spectral theorem for self-adjoint compact operators.
- Fourier transform on R^n and the Plancherel Theorem.
- Radon measures and the Riesz representation theorem for positive linear functionals.
After completing the course, the student will have:
Knowledge about the subjects mentioned in the description of the content.
Skills to solve problems concerning the material covered.
The following Competences:
- Have a good understanding of the fundamental concepts and results presented in lectures, including a thorough understanding of various proofs.
- Establish connections between various concepts and results, and use the results discussed in lecture for various applications.
- Be in control of the material discussed in the lectures to the extent of being able to solve problems concerning the material covered.
- Be prepared to work with abstract concepts (from analysis and measure theory).
- Handle complex problems concerning topics within the area of Functional Analysis.
Academic qualifications equivalent to a BSc degree is recommended.
- Theory exercises
- 7,5 ECTS
- Type of assessment
- Continuous assessmentTwo mandatory sets of exercises. Each set counts for 50% of the grade.
- All aids allowed
- Marking scale
- 7-point grading scale
- Censorship form
- External censorship
30 minutes oral examination with 30 minutes preparation with all aids.
Criteria for exam assesment
The student must in a satisfactory way demonstrate that he/she has mastered the learning outcome.