Introduction to Mathematical Analysis (7 ECTS)
Elements of set theory (finite, enumerable, uncountable sets). The E system od real numbers as a basic mathematical structures of a fully ranked body with the Archemedian quality. Real sequences, subsequences, upper and lower limits of a real sequence, convergence of a real sequence, real series and applications. The C system of complex numbers as a basic mathematical structure of a vector space with an internal product and a derived level (Euclidean space) .
The metric function and metric space definition. Metric spaces topology (neighbourhood of a point, internal and marginal points to given set, open and closed sets, metric subspace). Set diameter, point and set distance from a given set, bounded sets, compact sets, perfect sets, connected sets. Compact subsets of Euclidean metric Rn spaces. Point sequence convergence in a metric space, subsequences limits, the Cauchy condition, metric space completeness. Limit and continuity of a function from one metric space to another. Ηomeomorphism and isometry between metric spaces. Function continuity to compact sets and uniform continuity. Maximisation/ Minimisation of continuous functions, kurtosis and applications. Uniform convergence of a sequence of functions.
Introduction to vector space theory with metric derived from a level or internal product. Euclidean spaces examples, Banach spaces, Hilbert spaces. Graphical representations of vector spaces with levels, dual space. Introduction to the Riemann - Stieltjes and the Lebesque theory of integration.
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- W. Rudin (1974): Real and Complex Analysis, 2nd Edition, McGraw-Hill, Inc.