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German original

Functional Analysis
(dt. Funktionalanalysis)

Level, degree of commitment in original study programme Advanced 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
Translation missing. German original:
Studienleistung: Erreichen von mindestens 50 Prozent der Punkte aus den wöchentlich zu bearbeitenden Übungsaufgaben.
Prüfungsleistung: Klausur oder mündliche Prüfung
Language,
Grading
German,
The grading is done with 0 to 15 points according to the examination regulations for study course M.Sc. Mathematics.
Original study programme M.Sc. Mathematik / Vertiefungsbereich Mathematik
Duration,
frequency
One semester,
Regelmäßig im Wechsel mit anderen advanced moduleen im Gebiet Analysis
Person in charge of the module's outline Prof. Dr. Ilka Agricola, Prof. Dr. Stephan Dahlke, 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

The students shall

  • get to know typical problems of infinite-dimensional theory and their applications,
  • learn on examples like minimization problems how pure and applied mathematics interact,
  • practice mathematical work methods (development of mathematical intuition and its formal justification, training of the ability to abstract, proof techniques),
  • improve their oral communication skills in the exercises by practicing free speech in front of an audience and during discussion.

Prerequisites

Translation is missing. Here is the German original:

Keine. Empfohlen werden die Kompetenzen, die in den Basismodulen Analysis und Lineare Algebra sowie im Modul Maß- und Integrationstheorie vermittelt werden.


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 Wintersemester 2018/19. 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.