*Instructor:* Dr. Tamas TASNÁDI;

*Text:* class notes

*Reference books:*

— Walter Rudin: Principles of Mathematical Analysis (any edition)

— Miklós Laczkovich, Vera T. Sós: Real analysis, Foundations and Func- tions of One Variable, (Springer, 2015)

— Miklós Laczkovich, Vera T. Sós: Real analysis, Series, Functions of several Variables, and Applications (Springer, 2017)

*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 calcu- lus.
We do not follow closely a single textbook, but for those who wish to consult,
the relevant chapter numbers of the reference books will be given.

*Topics:*

- Review: Real numbers, numerical sequences.
- Differentiation I.: Limit, continuity and differentiation of single variable, real functions. Mean value theorems. Applications.
- Integration: Indefinite and Riemann integrals, inequalities, estimating sums with integrals.
- Metric spaces: Euclidean spaces, Topology, convergence, continuity, compactness, connectedness, completeness, separability. The metric space C([a,b]).
- Differentiation II.: Derivation of functions of several variables.
- Infinite series: Numerical series, sequences and series of functions, power series and analytic functions.