Chapters in higher mathematics

B232 - Summer 23/24

Chapters in higher mathematics - XP33CHM

Credits 4
Semesters Undefined
Completion Exam
Language of teaching English
Extent of teaching 2P
The course consists of several deeper results in a few mathematical disciplines. The idea is to help a student to read, with a certain comfort, the monographs in given lines of applied mathematics. The contents of the course are fundamental results (principles) of nowadays mathematics. More specifically, the course concerns the Stone representation theorem for Boolean algebras (as applied in mathematical logics and probability theory), the Banach fixed-point theorem for complete metric spaces (as applied in numerical mathematics), the Tychonoff theorem on compact spaces (as applied in measure theory), the Riesz representation theorem for linear forms in a Hilbert space (as applied in the optimization theory), the Brower theorem for balls in Rn (as applied in linear algebra – the Perron theorem), the elements of category theory for a practical man, etc. The asset may be a certain encouragement in a student’s research.
Course outlines
1. Introduction. Metric spaces
2. Connectedness and the curve connectedness in metric spaces
3. Compact metric spaces
4. Complete metric spaces and the Banach fixed-point theorem
5. Elementary proof of the Fundamental theorem of algebra
6. Lattices and Boolean algebras
7. The Stone representation for Boolean algebras
8. Extension of states on a Boolean algebra (the Tychonoff theorem)
9. Categories and morphisms
10. Normal and Hilbert spaces
11. The Riesz representation theorem for linear forms
12. The Sperner lemma
13. The Brower theorem on the fixed-points on the continuous mappings on ball in Rn
14. An application on Brower theorem: The Perron theorem on eigenvalues
Mandatory bibliography:
Hoggar, S. G.:Mathematics for computer graphics. Cambridge University Press, Cambridge, 1992.
Rudin, W.: Functional analysis. Second edition. McGraw-Hill, Inc., New York, 1991.
Recommended bibliography:
Rudin, W.: The Principles of Mathematical Analysis 3rd Edition. McGraw-Hill Publishing Company, 2006