Modul

Written examination of 120 minutes.

The students can

• explain basic topological concepts such as compactness in the framework of metric spaces, and are able to apply these in examples.
• describe the structure of Hilbert spaces and can use them in applications.
• explain the principle of uniform boundedness, the open mapping theorem and the Hahn-Banach theorem, and are able to derive conclusions from them.
• describe the concepts of dual Banach spaces, in particular weak convergence, reflexivity and the Banach-Alaoglu theorem. They can discuss these concepts in examples.
• explain the spectral theorem for compact self-adjoint operators.
• come up with a proof for simple functional analytic statements.

None

• Metric spaces (basic topological concepts, compactness),
• Hilbert spaces, Orthonormal bases, Sobolev spaces,
• Continuous linear operators on Banach spaces (principle of uniform boundedness, open mapping theorem),
• Dual spaces and representations, Hahn-Banach theorem, Banach-Alaoglu theorem, weak convergence, reflexivity,
• Spectral theorem for compact self-adjoint operators.

Total workload: 240 hours

Attendance: 90 h

• lectures, problem classes and examination

Self studies: 150 h

• follow-up and deepening of the course content,
• work on problem sheets,
• literature study and internet research on the course content,
• preparation for the module examination