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:


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