Matematiska strukturer

**Valid from:** Spring 2023**Decided by:** Maria Sandsten**Date of establishment:** 2023-01-14

**Division:** Mathematics**Course type:** Course given jointly for second and third cycle**The course is also given at second-cycle level with course code:** FMAN65**Teaching language:** Swedish

Besides mere knowledge imparting, the course aims to give training in theorem proving, and to bring out the possibilities of a more abstract representation of mathematical concepts and the connections between them. The intention is to give an overall view elucidating the foundations of the mathematical theories in the basic courses.

*Knowledge and Understanding*

For a passing grade the doctoral student must

- be familiar with and in his or her own words be able to explain the concepts within analysis, algebra and geometry touched upon in the course.
- be able to give examples of how these concepts are abstractions of concepts in the basic courses, and show understanding for how the abstractions serve to simplify and clarify the theory.
- in his/her own word be able to describe the logical connections between the concepts (theorems and proofs).

*Competences and Skills*

For a passing grade the doctoral student must

- be able to demonstrate ability to identify problems which can be modelled with the concepts introduced.
- in the context of problem solving be able to demonstrate ability to, in simple situations, develop the theory further.
- with proper terminology, in a well-structured manner, and with clear logic be able to explain the connections between various concepts in the course.
- with proper terminology, suitable notation, in a well-structured manner and with clear logic be able to explain the solution to a problem or the proof of a theorem.
- have developed his or her ability to independently read and judge mathematical text at a high level.

Sets. Real numbers. Metric spaces. Algebra (groups and linear spaces). Banach spaces and Hilbert spaces with applications.

Kaplansky, I.: Set Theory and Metric Spaces. American Mathematical Society. 2001. ISBN 9780821826942.

**Types of instruction:** Lectures, seminars

**Examination formats:** Written exam, oral exam.
The examiner, in consultation with Disability Support Services, may deviate from the regular form of examination in order to provide a permanently disabled student with a form of examination equivalent to that of a student without a disability.**Grading scale:** Failed, pass**Examiner:**

**Assumed prior knowledge:** FMAF01 Analytic functions and FMAF05 Systems and Transforms, or equivalent.

**Course coordinators:** **Web page:** https://canvas.education.lu.se/courses/20374