# MM543: Measure and Integration and Banach spaces

The Study Board for Science

Teaching language: Danish or English depending on the teacher, but English if international students are enrolled

EKA: N300037112, N300037102

Assessment: Second examiner: External, Second examiner: None

Grading: 7-point grading scale, Pass/Fail

Offered in: Odense

Offered in: Spring

Level: Bachelor

STADS ID (UVA): N300037101

ECTS value: 10

Date of Approval: 01-11-2022

Duration: 1 semester

Version: Archive

## Comment

## Entry requirements

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

## Academic preconditions

Material from MM533 Mathematical and Numerical Analysis (or MM535 Topology) together with MM505 Linear Algebra or MM538 Algebra and Linear Algebra or MM540 Mathematical Methods for Economics og MM568 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:

- apply the thinking and terminology from the subject's basic disciplines.
- analyze and evaluate the theoretical and practical problems for the application of a suitable mathematical model.

Give knowledge and understanding of:

- basic knowledge generation, theory and methods in mathematics.
- 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

## Content

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, the Radon-Nikodym theorem.

## Literature

## Examination regulations

## Exam element b)

## Timing

June

## Tests

## Oral exam

## EKA

N300037112

## Assessment

Second examiner: External

## Grading

7-point grading scale

## Identification

Student Identification Card

## Language

Normally, the same as teaching language

## Duration

30 minutes

## Examination aids

To be announced during the course

## ECTS value

5

## Exam element a)

## Timing

Spring

## Tests

## Mandatory assignment

## EKA

N300037102

## Assessment

Second examiner: None

## Grading

Pass/Fail

## Identification

Full name and SDU username

## Language

Normally, the same as teaching language

## Examination aids

To be announced during the course

## ECTS value

5

## Indicative number of lessons

## Teaching Method

At the faculty of science, teaching is organized after the three-phase model ie. intro, training and study phase.

- Intro phase (lectures) - 56 hours
- Training phase: 28 hours
- Study phase: 20 hours

Form of instruction:

- Reading of suggested literature
- Preparation of exercises in study groups

Educational activities: Teaching is centred on interaction and dialogue. In the intro phase, concepts, theories and models are introduced and put into perspective. In the training phase, students train their skills through exercises and dig deeper into the subject matter. In the study phase, students gain academic, personal and social experiences that consolidate and further develop their scientific proficiency. Focus is on immersion, understanding, and development of collaborative skills.

## Teacher responsible

## Timetable

## Administrative Unit

## Team at Educational Law & Registration

## Offered in

## Recommended course of study

## Transition rules

Transitional arrangements describe how a course replaces another course when changes are made to the course of study.

If a transitional arrangement has been made for a course, it will be stated in the list.

See transitional arrangements for all courses at the Faculty of Science.