Introduction to Mathematical Analysis (7 ECTS)

Introduction to Real Analysis. Fundamental concepts from set theory. The set of real numbers. Sequences and series of real numbers.

Real functions, continuous, uniformly continuous, monotone and convex functions. Stieltjes integral and functions of bounded variation. Metric spaces and continuous functions in metric spaces. Uniform convergence of sequences and series of functions. Linear spaces with norm and inner product spaces (Banach and Hilbert spaces). Short introduction to Lebesgue measure and integration. Applications of these concepts in probability, statistics and scientific computation.

      Recommended Reading

• K. Saxe, Beginning Functional Analysis, Springer Series on Undergraduate Mathematics, 2002
• A.N. Yannacopoulos, Introduction to Mathematical Analysis, Lecture Notes (2016)
• Johnsonbaugh, R. and W. Pfaffenberger (1981). Foundations of mathematical analysis. M. Dekker (New York, NY).
• Labarre, A. E. (2008). Intermediate mathematical analysis. Dover Publications
• Bobrowski, A. (2005). Functional analysis for probability and stochastic processes: an introduction. Cambridge University Press.
• Rudin, W. (1964). Principles of mathematical analysis, Volume 3. McGraw-Hill New York.
• Severini, T. A. (2005). Elements of distribution theory, Volume 17. Cambridge University Press.
• Jacod, J. and P. E. Protter (2003). Probability essentials. Springer.

The courses outline is here.