13844 - Functional Analysis Modulübersicht

Module Number: 13844
Module Title:Functional Analysis
  Funktionalanalysis
Department: Faculty 1 - Mathematics, Computer Science, Physics, Electrical Engineering and Information Technology
Responsible Staff Member:
  • Prof. Dr. rer. nat. habil. Wachsmuth, Gerd
Language of Teaching / Examination:English
Duration:1 semester
Frequency of Offer: On special announcement
Credits: 8
Learning Outcome:The students have
  • expanded and intensified their knowledge from previous modules of Analysis and Algebra
  • competently mastered definitions and interrelations within abstract spaces
  • become acquainted with applications in Numerics, Optimization, and Physics
  • acquired basic knowledge for advanced modules
  • became familiar with fundamental techniques of proof
  • improved their logical way of thinking by solving problems in abstract spaces
  • further developed their abilities for independent scientific work by treating themes from Functional Analysis
Contents:
  • Normed spaces
    completion, separable spaces, Lebesgue spaces, spaces of continuous and differentiable functions, Sobolev spaces
  • Linear and continuous operators
    Projection and adjoint operators, topological dual spaces, completely continuous operators, weak convergence and reflexivity
  • Main theorems
    Weierstrass, Hahn-Banach, Schauder, the openmapping, the closed graph
  • Hilbert spaces
    Spectral theorem for selfadjoint, completely continuous operators
Recommended Prerequisites:Knowledge of the content of the modules
  • 11103 Analysis I
  • 11104 Analysis II
  • 11201 Analysis III
Mandatory Prerequisites:No successful participation in module 11303 - Funktionalanalysis.
Forms of Teaching and Proportion:
  • Lecture / 4 Hours per Week per Semester
  • Exercise / 2 Hours per Week per Semester
  • Self organised studies / 150 Hours
Teaching Materials and Literature:
  • Aubin, J.-P.: Applied Functional Analysis, Wiley, 2000, https://doi.org/10.1002/9781118032725
  • Brezis, H.:  Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011, https://doi.org/10.1007/978-0-387-70914-7
  • Rudin, W.: Functional Analysis, McGraw Hill, 1991
Module Examination:Prerequisite + Final Module Examination (MAP)
Assessment Mode for Module Examination:Prerequisite:
  • Successful completion of homework
Final module examination:
  • Written examination, 90 min. OR
  • Oral examination, 30 min. (with small number of participants)
In the first lecture it will introduced, if the examination will organized in written or oral form.
Evaluation of Module Examination:Performance Verification – graded
Limited Number of Participants:None
Part of the Study Programme:
  • Master (research-oriented) / Angewandte Mathematik / PO 2019
  • Master (research-oriented) / Artificial Intelligence / PO 2022
  • Bachelor (research-oriented) / Mathematik / PO 2019
  • Bachelor (research-oriented) / Mathematik / PO 2023
  • Bachelor (research-oriented) - Co-Op Programme with Practical Placement / Mathematik - dual / PO 2023
  • Master (research-oriented) / Physics / PO 2021
  • Bachelor (research-oriented) / Physik / PO 2021
  • Bachelor (research-oriented) / Wirtschaftsmathematik / PO 2023
  • Bachelor (research-oriented) - Co-Op Programme with Practical Placement / Wirtschaftsmathematik - dual / PO 2023
Remarks:
  • Study programme Angewandte Mathematik M.Sc.: Compulsory elective module in complex „Analysis / Algebra / Kombinatorik“
  • Study programme Mathematik B.Sc.: Compulsory elective module in complex „Vertiefung“, in limited extend 
  • Study programme Wirtschaftsmathematik B.Sc.: Compulsory elective module in complex „Vertiefung“, in limited extend
  • Study programme Physics M.Sc.: Compulsory elective module in complex „Minor Subject“
  • Study programme Artificial Intelligence M.Sc.: Compulsory elective module in complex  „Advanded Methods“
If there is no need that the module is taught in English, alternatively the german version 11303 „Funktionalanalysis“ may beread instead.
Module Components:
  • Lecture: Functional Analysis
  • Accompanying exercises
  • Related examination
Components to be offered in the Current Semester: