Introduction to Mathematical Analysis (7 ECTS)

Course Code: 
Elective Courses

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. 

      Recommended Reding

  • Rudin, W. (2000): Αρχές Μαθηματικής Αναλύσεως, Εκδόσεις Leader Books, Αθήνα.
  • Καρυοφύλλης, X.  (1995): Στοιχεία Συναρτησιακής Ανάλυσης, Εκδόσεις Ζήτη, Θεσσαλονίκη.
  • Τ. Μ. Αpostol (1974): Mathematical Analysis, 2nd  Edition, Addison-Wesley Publishing Company, Inc.
  • A. N. Kolmogorov and S. V. Fomin (1975): Introductory Real Analysis, Dover Publications, Inc.
  • H. L. Royden (1968): Real Analysis, 2nd Edition, Macmillan Publishing Company, Inc.
  • W. Rudin (1974): Real and Complex Analysis, 2nd Edition, McGraw-Hill, Inc.