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Functional Analysis (Winter Semester 2009/10)

  • Lecturer: Prof. Dr. Roland Schnaubelt
  • Classes: Lecture (1048), Problem class (1049)
  • Weekly hours: 4+2
  • Audience: Mathematics, Physics (from 5. semester)
Schedule
Lecture: Tuesday 9:45-11:15 Nusselt-Hörsaal Begin: 20.10.2009
Thursday 11:30-13:00 Hertz-Hörsaal
Problem class: Friday 14:00-15:30 Eiermann Begin: 23.10.2009
Lecturers
Lecturer Prof. Dr. Roland Schnaubelt
Office hours: Wednesday, 11:30 - 13:00, and by appointment
Room 2-047 (Englerstr. 2) Kollegiengebäude Mathematik (20.30)
Email: schnaubelt@kit.edu
Problem classes Dr. Esther Bleich
Office hours: Nach Vereinbarung
Room 3A-28 Allianz-Gebäude (05.20)
Email: esther.bleich@t-online.de

The lecture is concerned with Banach and Hilbert spaces as well as linear operators acting on these spaces. Typical examples are spaces of continuous and integrable functions and linear maps, which one defines via integration of such functions. In this way one can formulate integral equations as affine or linear equations on a suitable Banach space, and one can solve them by means of functional analytic methods. This class of problems was in fact the historical starting point for the development of functional analysis around 1900. In the following years it became a fundamental area of modern analysis and its applications in- and outside of mathematics.

A preliminary list of topics:

  • basic properties and examples of metric and Banach spaces and of linear operators
  • Hilbert spaces (scalar product, orthogonal projection and basis)
  • principle of uniform boundedness and open mapping theorem
  • dual spaces and Theorem of Hahn-Banach
  • weak convergence and Theorem of Banach-Alaouglu
  • Fourier transform and applications to partial differential equations

Prerequisites: Analysis 1-3 and Linear Algebra 1+2.

Further informations concerning this lecture you find in the Studierendenportal of the KIT at the URL
https://studium.kit.edu/sites/vab/60678/Start/homepage.aspx

References

  • H.W. Alt: Lineare Funktionalanalysis. Springer.
  • J.B. Conway: A Course in Functional Analysis. Springer.
  • M. Schechter: Principles of Functional Analysis. Academic Press.
  • A.E. Taylor, D.C. Lay: Introduction to Functional Analysis. Wiley.
  • D. Werner: Funktionalanalysis. Springer.

(More literature can be found in the "Vorlesungspräsenz" in the department's library.)