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 - 1. SÄ 2021
  • Master (research-oriented) / Artificial Intelligence / PO 2022 - 1. SÄ 2024
  • Master (research-oriented) / Mathematical Data Science / PO 2025
  • Master (research-oriented) / Mathematics / PO 2025
  • 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“
  • Study programme Mathematics  M.Sc.: Compulsory elective module in complex „Analysis / Algebra / Combinatorics“
  • Study programme Mathematical Data Science  M.Sc.: Compulsory elective module in complex „Fundamentals of Data Science“
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:
  • no assignment