Functional analysis

Course goals

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Content

The course Functional Analysis is an elective for mathematics students. The course is prerequisite for students who want to specialize in analysis and the theory of partial differential equations. Please find more information on the students website.

Course content
The following subjects will be covered:

• (Semi-)norms, Banach spaces:
  • L^p-space
  • completion of a normed vector space
  • noncompactness of the closed unit ball in infinite dimensions
• Inner products, Hilbert spaces:
  • Cauchy-Schwarz inequality
  • parallelogram rule
  • orthonormal bases
  • Parseval's identity
  • Bessel's inequality
  • classification of Hilbert spaces
  • separability
  • Fourier series
• Linear operators:
  • bounded linear maps
  • Neumann series
  • Open Mapping Theorem (Banach-Schauder)
  • Closed Graph Theorem
  • Uniform Boundedness Theorem (Banach-Steinhaus)
• Dual spaces:
  • Fréchet-Riesz representation theorem
  • Hahn-Banach Theorem
  • bidual space and reflexivity
  • dual operator
• Operators on Hilbert spaces:
  • adjoint of an operator
  • spectrum
  • spectral radius
  • spectra of self-adjoint, unitary, normal, and compact operators
  • Riesz-Schauder Theorem about a compact perturbation of the identity operator
  • Spectral Theorem for a compact normal operator

Course format
Per week, there is two times a lecture of two hours and two times an exercise class of two hours.

Examination
A final exam takes place at the end of the course. The hand-in assignments also count toward the final grade. 

The final grade C for this course is calculated as follows: C = max(M,min((M+I)/2,M+1))
where I = hand-ins and M = max(T,H) with T = exam
and H = retake exam (i.e., the hand-ins are voluntary and can only count positively).

Retake and participation obligation
Participating in the retake is possible only either if C is at least 4.0, or if you have handed at least 4 hand-ins with non-zero grade.