Module Handbook

  • Dynamischer Default-Fachbereich geändert auf MAT

Course MAT-70-11-K-4

Functional Analysis (4V+2U, 9.0 LP)

Course Type

SWS Type Course Form CP (Effort) Presence-Time / Self-Study
- K Lecture with exercise classes (V/U)
4 V Lecture 6.0 CP 56 h 124 h
2 U Exercise class (in small groups) 3.0 CP 28 h 62 h
(4V+2U) 9.0 CP 84 h 186 h

Basedata

SWS 4V+2U
CP, Effort 9.0 CP = 270 h
Position of the semester 1 Sem. in SuSe
Level [4] Bachelor (Specialization)
Language [EN] English
Lecturers
+ further Lecturers of the department Mathematics
Area of study [MAT-SPAS] Analysis and Stochastics
Livecycle-State [NORM] Active

Possible Study achievement

  • Verification of study performance: proof of successful participation in the exercise classes (ungraded)
  • Examination number (Study achievement): 84020 ("Exercise Class Functional Analysis")
  • Details of the examination (type, duration, criteria) will be announced at the beginning of the course.

Contents

  • Hahn-Banach theorem and its applications,
  • Baire category theorem and its applications (uniform boundedness principle, Banach-Steinhaus-Theorem, open mapping theorem, inverse mapping theorem, closed graph theorem),
  • Weak convergence (Banach-Alaoglu theorem, reflexive Banach spaces, Mazur lemma and its applications),
  • Projections (closed complement theorem),
  • Bounded operators (adjoint operator, spectrum, resolvent, normal operators),
  • Compact operators (Fredholm operators, Fredholm alternative and its applications, spectral theorem(Riesz-Schauder) and its application to normal operators),
  • Unbounded operators (graph, symmetric and self-adjoint operators).

Competencies / intended learning achievements

The students have studied and understand the mathematical concepts in infinite-dimensional spaces with a special emphasis on analytical aspects. They have gained the basic analytical tools required for solving differential and integral equations in theory and application.

Literature

  • H.-W. Alt: Lineare Funktionalanalysis,
  • H. Heuser: Funktionalanalysis,
  • M. Reed, M, B. Simon: Functional Analysis I,
  • D. Werner: Funktionalanalysis.

Materials

Further literature will be announced in the lecture; Exercise material is provided.

Registration

Registration for the exercise classes via the online administration system URM (https://urm.mathematik.uni-kl.de).

References to Course [MAT-70-11-K-4]

Module Name Context
[MAT-30-10L-M-5] Specialisation Module (Teachers Training Programme Mathematics) WP: Obligation to choose in Obligatory-Modulteil #A (Lectures) 4V, 6.0 LP
[MAT-70-11-M-4] Functional Analysis P: Obligatory 4V+2U, 9.0 LP
Course-Pool Name
[MAT-70-4V-KPOOL-4] Elective Courses Analysis and Stochastics (4V, B.Sc.)
[MAT-70-KPOOL-4] Specialisation Analysis and Stochastics (B.Sc.)
[MAT-80-4V-KPOOL-4] Elective Courses Modelling and Scientific Computing (4V, B.Sc.)