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Functional Analysis
(dt. Funktionalanalysis)

Level, degree of commitment Specialization module, compulsory elective module
Forms of teaching and learning,
workload		 Lecture (4 SWS), recitation class (2 SWS),
270 hours (90 h attendance, 180 h private study)
Credit points,
formal requirements		 9 CP
Course requirement(s): Successful completion of at least 50 percent of the points from the weekly exercises.
Examination type: Written or oral examination (individual examination)
Language,
Grading		 English,
The grading is done with 0 to 15 points according to the examination regulations for the degree program M.Sc. Mathematics.
Duration,
frequency		 One semester,
Regularly alternating with other specialization modules im Gebiet Analysis
Person in charge of the module's outline Prof. Dr. Ilka Agricola, Prof. Dr. Pablo Ramacher

Contents

  • Banach and Hilbert spaces, their dual spaces
  • strong and weak convergence, pre-compactness, convex sets and minimization problems
  • continuous operators, dual operators, operator topologies, Fourier and Laplace transformations
  • Standard theorems of functional analysis
  • Spectrum of bounded operators, Fredholm alternative, Fredholm operators and their index, spectral decomposition of normal operators
  • Unbounded operators: basic questions, differential operators

Qualification Goals

Students will

  • know typical problems of infinite-dimensional theory and their applications,
  • understand the close interlocking of pure and applied mathematics through examples such as minimization problems,
  • have deepened mathematical working methods (developing mathematical intuition and its formal justification, abstraction, proof),
  • have improved their oral communication skills in exercises by practicing free speech in front of an audience and in discussion.

Prerequisites

None. The competences taught in the following modules are recommended: either Linear Algebra I and Linear Algebra II or Basic Linear Algebra, either Analysis I and Analysis II or Basic Real Analysis, Measure and Integration Theory.


Applicability

The module can be attended at FB12 in study program(s)

  • B.Sc. Mathematics
  • B.Sc. Business Mathematics
  • M.Sc. Computer Science
  • M.Sc. Mathematics
  • M.Sc. Business Mathematics
  • LAaG Mathematics

When studying M.Sc. Mathematics, this module can be attended in the study area Compulsory Elective Modules in Mathematics.

The module can also be used in other study programs (export module).

The module is assigned to both Applied and Pure Mathematics. Further information on eligibility can be found in the description of the study area.


Recommended Reading

  • Friedrich Hirzebruch, Winfried Scharlau, Einführung in die
  • Funktionalanalysis. BI-Wissenschaftsverlag, 1991.
  • John B. Conway, A course in functional analysis. Springer-Verlag, 1990.
  • Walter Rudin, Functional analysis. McGraw-Hill, 1991.



Please note:

This page describes a module according to the latest valid module guide in Winter semester 2023/24. Most rules valid for a module are not covered by the examination regulations and can therefore be updated on a semesterly basis. The following versions are available in the online module guide:

The module guide contains all modules, independent of the current event offer. Please compare the current course catalogue in Marvin.

The information in this online module guide was created automatically. Legally binding is only the information in the examination regulations (Prüfungsordnung). If you notice any discrepancies or errors, we would be grateful for any advice.