Reële Analyse

This course offers a rigorous introduction to analysis in metric spaces, a framework that extends the familiar geometry of Euclidean space to far more general settings with applications accross various fields of mathematics. Typical examples are:

• The space of continuous functions (Differential equations)

• The space of square integrable functions (Partial differential equations, Numerical analysis, Probability theory, Quantum mechanics, Graph theory, etc.)

• The space of summable sequences (Machine learning)

Topics treated during the course include:

• Cauchy and convergent sequences

• Open and closed sets

• (Uniform) continuity and homeomorphisms

• Total boundedness

• Completeness and fixed points

• Compactness

• Uniform convergence

All discussed topics culminate in a discussion of Fourier series for periodic continuous functions.