*Instructor:* Dr. Bence BORDA;

*Reference book:* Walter Rudin: Principles of Mathematical Analysis (any edition)

*Prerequisite:* Calculus

Course description: The most important concepts,
methods and applications of real analysis and the theory of metric spaces are covered,
with an emphasis on examples and problem solving.
The course is self-contained, the only prerequisite is calculus.
We do not follow any textbook, but for those who wish to consult one Rudin's Principles of Mathematical Analysis is recommended.

*Topics:*

- Review: sets, functions, real numbers, Euclidean spaces
- Metric spaces: topology, convergence, continuity, compactness, connectedness, completeness, separability
- Differentiation: derivatives of functions of one or several variables, mean value theorems, minimum and maximum of functions
- Integrals: indefinite and Riemann integrals, inequalities, estimating sums with integrals
- Infinite series: numerical series, sequences and series of functions, the metric space C([a,b]), power series and analytic functions