Functional analysis

Vorlesung in englischer Sprache im Wintersemester 2002/2003 an der TU Chemnitz.

Coordinates:
Lectures and exercise classes: Do 1. 2/N 001, Fr 2. 2/B102

Introduction: Functional analysis is a field that has a central position between applied and pure fields that deal in one way or the other with function spaces.
Of course, a thorough understanding of basic analysis and linear algebra is necessary to follow the course. There is no text book for the lecture but literature for complementary reading will be posted below.
The exercises will consist of two different components: Exercise sheets that students should work on on their own and working sections that will be embedded in the regular course:

Content:
Introduction
0. Banach spaces: the first encounter
1. Basic structures
  1.1 Linear algebra
  1.2 Metric spaces and topology
  Working section A: nets and convergence
  1.3 Norms and scalar products
  1.4 Linear operators
Working section B: Finite vs infinite dimensions
Working section B: solution proposed by K. Luther and D. Oriwol
2. The Hahn-Banach theorem
  2.1 The Hahn-Banach extension theorem
  2.2 The Hahn-Banach separation theorem
  2.3 The bipolar theorem
Working section C: subspaces, quotients and their duals
3. Baire's theorem and its consequences
  3.1 Baire's theorem
  3.2 The open mapping theorem and Banach's isomorphism theorem
  3.3 The closed graph theorem
  3.4 The uniform boundedness principle
  3.5 The Banach-Steinhaus theorem
4. Dual spaces and adjoints
  4.1 Examples of dual spaces
  4.2 Adjoint operators and the closed range theorem
Working section D: completion of normed, metric and pre-Hilbert spaces
5. More spaces
  5.1 Hilbert spaces: basic geometry
  5.2 Spaces of continuous functions

To be continued ...

Any questions? The answer is ... here

Literature:

• Functional Analysis
      K. Yosida
      Springer, Berlin 1968
  
• Funktionalanalysis: Ein Arbeitsbuch
      M. Mathieu
      Spektrum Akademischer Verlag, Heidelberg 1998
  
• Lineare Funktionalanalysis: Eine anwendungsorientierte Einführung
      H.W. Alt
      Springer, Berlin 1992
  
• Einführung in die Funktionalanalysis
      R. Meise, D. Voigt
      Vieweg, Braunschweig 1992