312.103 (17W) Funktionalanalysis

Wintersemester 2017/18

Anmeldefrist abgelaufen.

Erster Termin der LV
03.10.2017 08:00 - 10:00 , N.2.01
... keine weiteren Termine bekannt

Überblick

Lehrende/r
LV-Titel englisch
Functional Analysis
LV-Art
Vorlesung-Übung (prüfungsimmanente LV )
Semesterstunde/n
2.0
ECTS-Anrechungspunkte
3.0
Anmeldungen
11 (25 max.)
Organisationseinheit
Unterrichtssprache
es wurde keine Unterrichtssprache angegeben
LV-Beginn
01.10.2017

LV-Beschreibung

Intendierte Lernergebnisse

Understanding of strongly continuous semigroups.  

Lehrmethodik

Lecture

Inhalt/e

Initial point of the course is Cauchy’s functional equation, i.e. the problem to classify those functions T(t), which satisfy the semigroup property

T(t+s)=T(t)T(s) for all nonnegative t,s

This seemingly theoretical question has not only far reaching applications in ordinary, partial and functional differential equations, but also requires to develop an interesting functional analytical machinery. Among these tools are closed operators and their spectrum (which extends the rather trivial results from Linear Algebra), the important special case of compact operators, linear semigroups and their generators. Moreover, we provide a modern application to time-dependent ordinary differential equations in terms of evolution families. 

Contents

  • Spectral theory for closed operators
  • Riesz-Schauder theory for the spectrum of compact operators
  • Linear semigroups and their generators
  • Hille-Yosida theorem characterising the solutions to Cauchy’s equation
  • Evolution families


Erwartete Vorkenntnisse

Analysis, Linear Algebra, Ordinary differential equations, Basics on Functional Analysis, some knowledge on evolutionary PDEs won't hurt

Literatur

K.J. Engel & R. Nagel: One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics 194, Springer, Berlin, 2000

A. Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Springer, Berlin, 1983

K. Yosida: Functional Analysis, Grundlehren der Mathematischen Wissenschaften 123, Springer, Berlin, 1980

Prüfungsinformationen

Prüfungsmethode/n

oral examination (45 min) after the last lecture, by appointment

Prüfungsinhalt/e

contents of the course

Beurteilungskriterien/-maßstäbe

familiarity with the contents of the course

Beurteilungsschema

Note/Grade Benotungsschema

Position im Curriculum

  • Doktoratsprogramm Modeling-Analysis-Optimization of discrete, continuous and stochastic systems (SKZ: ---, Version: 16W.1)
    • Fach: Modeling-Analysis-Optimization of discrete, continuous and stochastic systems (Pflichtfach)
      • Modeling-Analysis - Optimization of discrete, continuous and stochastic systems ( 0.0h XX / 0.0 ECTS)
        • 312.103 Funktionalanalysis (2.0h VU / 3.0 ECTS)
  • Masterstudium Technische Mathematik (SKZ: 401, Version: 13W.1)
    • Fach: Analysis (Pflichtfach)
      • Funktionalanalysis ( 2.0h VU / 3.0 ECTS)
        • 312.103 Funktionalanalysis (2.0h VU / 3.0 ECTS)

Gleichwertige Lehrveranstaltungen im Sinne der Prüfungsantrittszählung

Diese Lehrveranstaltung ist keiner Kette zugeordnet