Real numbers. Metric spaces. Basic topological concepts. Convergence. Continuity.
Derivative. Integral. Uniform convergence. Spaces of functions.
Banach´s fixed point theorem. Implicit and inverse mapping theorem. (Something about Lebesgue integral, alternatively something about differential forms and Stokes theorem.)
The course is a fundamental course for studies in more advanced mathematics and for studies in closely related fields.
By the end of the course the student should be able to solve problems on the different topics of the course. In particular the student should be able to
- Understand and be able to apply basic topological concepts. Be able to state the theorems of Heine-Borel and Bolzano-Weierstrass.
- Understand and be able to apply the concepts of continuity, convergence and derivative for functions between metric spaces. Be able to state Arzelà-Ascoli´s theorem and Weierstrass´ approximation theorem.