Reële Analyse

The course consists of two parts.

The first part of the course is about metric spaces. Metric spaces are a generalization of, e.g., the Euclidean space and are needed in applications to various other mathematical fields. Common 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)

You will learn about metric spaces during the course. Among other things, we will discuss: convergent sequences, open and closed sets, continuity and homeomorphisms, total boundedness and completeness, compactness, uniform convergence and spaces of continuous functions.

The second part of the course is about measure and integration theory. This theory is at the foundation of a large part of modern mathematics. It is used in Fourier analysis, (partial) differential equations, probability theory and numerical analysis. In measure theory, we study how to measure a set in a fair way. We will be able calculate volumes and areas of sets that cannot be treated with the Riemann integration theory from Analysis 1 and 2. This will lead us to a larger class of functions which we can integrate and very strong convergence theorems. We will end the second part of the course with an application to Fourier series.