NMAK10008U Functional Analysis (FunkAn)

Volume 2023/2024
Education

MSc Programme in Mathematics
MSc Programme in Statistics
MSc Programme in Mathematics with a minor subject
MSc Programme in Quantum Information Science

Content

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.
Learning Outcome

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.

Analyse 0 (An0), Analyse 1 (An1), Analyse 2 (An2) or Lebesgueintegralet og målteori (LIM), Topology (Top) and AdVec.

Academic qualifications equivalent to a BSc degree is recommended.
5 hours lectures (2+2+1) and 3 hours of exercises per week for 8 weeks.
• Category
• Hours
• Lectures
• 40
• Preparation
• 116
• Theory exercises
• 24
• Exam
• 26
• Total
• 206
Continuous feedback during the course of the semester
Credit
7,5 ECTS
Type of assessment
Written examination, 3 hours under invigilation
Continuous assessment
Type of assessment details
Two written assignments and a Final 3 hours written exam, under invigilation. Each of the written assignments count 25% towards the final grade; the students will be given 5 days to work on each. The Final written exam counts 50% towards the final grade, and it takes place in week 9.
Aid
All aids allowed
Marking scale