Syllabus academic year 2007/2008
 MATHEMATICAL STRUCTURES FMA111

Higher education credits: 6. Grading scale: TH. Level: A (Second level). Language of instruction: The course will be given in Swedish. FMA111 overlap following cours/es: FMA110 och FMA110. Compulsory for: Pi3. Optional for: D3, E3, F3, F3tmb. Course coordinator: Director of Studies, Lars-Christer Böiers, Lars_Christer.Boiers@math.lth.se, Matematik. Recommended prerequisits: FMA037 Complex analysis and FMA036 Linear analysis. Assessment: Written and/or oral test, to be decided by the examiner. Home page: http://www.maths.lth.se/matematiklth/vitahyllan/vitahyllan.html.

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 the concepts and their connections. 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 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 of 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).

Skills and abilities
For a passing grade the student must

be able to show capability to identify problems which can be modelled with the concepts introduced.

in the context of problem solving be able to show capability to, in simple situations, develop the theory further.

with proper terminology, well structured and with clear logic be able to explain the connections between various concepts in the course.

with proper terminology, suitable notation, well structured 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 independently to read and judge mathematical text at a high level.

Contents
Sets. Real numbers. Topology (metric spaces and general topological spaces). Algebra (groups and linear spaces). Banach spaces and Hilbert spaces with applications.

Literature
Fontes, M: Matematiska strukturer. Matematikcentrum 2006.