MM548: Measure and Integration and Banach spaces (7.5 ECTS)

STADS: 13017201

Level
Bachelor course

Teaching period
The course is offered in the spring semester.

Teacher responsible
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Timetable
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Prerequisites:
None

Academic preconditions:
Material from MM533 Mathematical and Numerical Analysis (or MM535 Topology) together with MM505 Linear Algebra or MM538 Algebra and Linear Algebra should be known.

Course introduction
The aim of the course is 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. The course builds on the knowledge acquired in the course Mathematical & Numerical Analysis, and gives an academic basis for studying Probability, that is part of the degree.

In relation to the competence profile of the degree it is the explicit focus of the course to:

• Give the competence to handle complex situations in study.
• Give skills to:
1. apply the thinking and terminology from the subject's basic disciplines.
2. analyze and evaluate the theoretical and practical problems for the application of a suitable mathematical model.
• Give knowledge and understanding of:
1. basic knowledge generation, theory and methods in mathematics.
2. how to conduct analyses using mathematical methods and critically evaluate scientific theories and models.

Expected learning outcome
The learning objective of the course is that the student demonstrates the ability 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
• 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
The following main topics are contained in the course: 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.

Literature
There isn't any litterature for the course at the moment.

Website
This course uses e-learn (blackboard).

Prerequisites for participating in the exam
None

Assessment and marking:

1. a. Mandatory assignment. Evaluated on a pass/fail basis by lecturer (5 ECTS). (13017202). 
2. Oral exam. Evaluated by external marking by the Danish 7-mark scale (2,5 ECTS). (13017212). 

Expected working hours
The teaching method is based on three phase model.
Intro phase: 42 hours
Skills training phase: 22 hours, hereof:
 - Tutorials: 22 hours

Educational activities Study phase: 15 hours

Educational form


Language
This course is taught in Danish or English, depending on the lecturer. However, if international students participate, the teaching language will always be English.

Remarks
The course is co-read with: MM517 Measure and Integration, and MM543 Measure and Integration and Banach Spaces.

The course cannot be chosen by students who took: MM517 Measure and Integration, or MM514 Hilbert and Banach Spaces.

Course enrollment
See deadline of enrolment.

Tuition fees for single courses
See fees for single courses.