MM543: Measure and Integration and Banach spaces (10 ECTS)

STADS: 13012501

Level
Bachelor course

Teaching period
The course is offered in the spring semester.

Teacher responsible
Email: szymanski@imada.sdu.dk

Timetable
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Comment:
Ubegrænset deltagerantal. Fælles undervisning med MM517 Mål- og integralteori (1.del) og MM514 Hilbert-og Banachrum (2.del)

Prerequisites:
None

Academic preconditions:
Material from MM533 Mathematical and Numerical Analysis (or MM535 Topology) together with MM505 Linear Algebra should be known.

Course introduction
To introduce measure and integration theory as well as modern functional analysis focused on Hilbert and Banach space theory. Moreover, this course lays foundation for further study of probability theory.

Qualifications
At the conclusion of the course the student should be able to:

• use methods from the theory to solve practical problems, especially related to the integral convergence theorems, Fubini’s theorem, and calculate Fourier series
• present statements and proofs of items from an a priori given list of topics
• Answer additional questions from teachers and examiners around the central concepts and results from the above list

Expected learning outcome
At the conclusion of the course the student should be able to:

• Understand the basic theory in this area, including in particular concepts of sigma-algebra, measurability, integral
• use methods from the theory to solve problems related to measure and integration and Banach space theory
• give an oral and written presentation in correct mathematical language

Subject overview
sigma-algebras, measurable maps, measure and integration with respect to measure, the Lebegue measure on the real line and on Rn, product measure, Lp-spaces, Hilbert spaces, Fourier series in Hilbert space, the projection theorem, introductory Banach space theory, the open mapping theorem and the Hahn-Banach extension theorem.

Literature

Meddeles ved kursets start.

Website
This course uses e-learn (blackboard).

Prerequisites for participating in the exam
None

Assessment and marking:

1. Mandatory assignment 1. Evaluated on a pass/fail basis by lecturer (0 ECTS). (13012512)
2. Mandatory assignment 2. Evaluated on a pass/fail basis by lecturer (0 ECTS). (13012522)
3. Oral exam. Evaluated by external censorship by the Danish 7-mark scale (10 ECTS). (13012502)

Expected working hours
The teaching method is based on three phase model.
Intro phase: 56 hours
Skills training phase: 28 hours, hereof:
 - Tutorials: 28 hours

Educational activities Study phase: 20 hours

Language
This course is taught in Danish or English, depending on the lecturer.

Course enrollment
See deadline of enrolment.

Tuition fees for single courses
See fees for single courses.