MM815: Hilbert- and Banach spaces (5 ECTS)

STADS: 13009701

Level
Master's level course

Teaching period
The course is offered in the autumn semester.

Teacher responsible
Email: dkyed@imada.sdu.dk

Additional teachers
szymanski@imada.sdu.dk
barlak@imada.sdu.dk

Timetable
There is no timetable available for the chosen semester.

Comment:
Ubegrænset deltagerantal. Fælles undervisning med MM514.

Prerequisites:
None

Academic preconditions:
The content of the courses FF502 (or MM501 Calculus I, MM502 Calculus II), MM505 Linear Algebra, MM533 Mathematic and numerical analysis, MM535 Topology (or MM508 Topology I, MM509 Topology II), and MM517 Measure and Integration Theory must be known.

Course introduction
To introduce Functional Analysis with special emphasis on Hilbert- and Banach Space Theory. This gives the basics for more advanced studies in modern Functional Analysis, in particular in Operator Algebra Theory and Banach Space Theory.

Expected learning outcome
At the end of the course the student will be able to:

  • reproduce the definitions within the syllabus of the course in a precise mathematical language
  • reproduce results, together with complete proofs, within the syllabus of the course
  • relate the results within the syllabus of the course to each other
  • apply the theory to solve problems within the scope of the syllabus of the course 
  • compute Fourier series and Fourier integrals of functions and predict their convergence properties based on regularity properties of the functions
  • discuss the applications of functional analysis in the theory of differential equations
Subject overview
Hilbert spaces. Fourier series and Fourier integrals. Introduction to Banach Space Theory. The Banach-Steinhaus theorem. The open mapping theorem. The Hahn-Banach extension theorem. The representation theorem of Riesz for positive linear functionals on C(K).

Literature
    Meddeles ved kursets start.


Website
This course uses e-learn (blackboard).

Prerequisites for participating in the exam
Prerequisite test consisting of mandatory exercises. The exercises must be passed in order to take the oral exam. (13009712)

Assessment and marking:
Oral examination, grades according to the Danish 7-point scale and external censorship. (13009702)

Reexamination i the same exam period or immediately thereafter.

Expected working hours
The teaching method is based on three phase model.
Intro phase: 28 hours
Skills training phase: 14 hours

Educational activities Study phase: 10 hours

Language
This course is taught in English.

Course enrollment
See deadline of enrolment.

Tuition fees for single courses
See fees for single courses.