312.103 (17W) Functional Analysis

Wintersemester 2017/18

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Erster Termin der LV
03.10.2017 08:00 - 10:00 , N.2.01
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Course title german
Lecture - Practical class (continuous assessment course )
Hours per Week
11 (25 max.)
Organisational Unit
Language of Instruction
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Course Information

Learning Outcome

Understanding of strongly continuous semigroups.  



Course overview

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. 


  • 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

Prior knowledge

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


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

Exam information

Exam methodology

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

Exam topics

contents of the course

Exam mode

familiarity with the contents of the course

Grading scheme

Note/Grade Benotungsschema

Degree programmes

  • Thematic Doctoral Programme Modeling-Analysis-Optimization of discrete, continuous and stochastic systems (SKZ: ---, Version: 16W.1)
    • Fach: Modeling-Analysis-Optimization of discrete, continuous and stochastic systems (Compulsory subject)
      • Modeling-Analysis - Optimization of discrete, continuous and stochastic systems ( 0.0h XX / 0.0 ECTS)
        • 312.103 Functional Analysis (2.0h VU / 3.0 ECTS)
  • Master's degree programme Technical Mathematics (SKZ: 401, Version: 13W.1)
    • Fach: Analysis (Compulsory subject)
      • Funktionalanalysis ( 2.0h VU / 3.0 ECTS)
        • 312.103 Functional Analysis (2.0h VU / 3.0 ECTS)

Equivalent Courses for counting the exam attempts

This Course has no equivalent Courses