Mathematical Structures
Matematiska strukturer

FMAN65F, 6 credits

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

General Information

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

Aim

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.

Goals

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.

Course Contents

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

Course Literature

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

Instruction Details

Types of instruction: Lectures, seminars

Examination Details

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:

Admission Details

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

Course Occasion Information

Contact and Other Information

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