Real Analyses

Course Description

Topics include real and complex number systems, elements of point-set topology of Euclidean space, numerical sequences and series, continuity and differentiability for functions of one and several variables. The Riemann-Stieltjes integral as a generalization of the Riemann integral, sequences and series of functions, and Fourier series will be studied as well.

Learning Outcomes

  • Evaluate ordered sets, real and complex fields
  • Apply main concepts of metric spaces and their topology to concrete problems
  • Interpret the connectedness, compactness and Weierstrass theorem
  • Evaluate the Cauchy sequences, completeness of real numbers, convergence and absolute convergence of series
  • Interpret concepts of limits, continuity of functions in metric spaces, and their properties, differentiation and integration
  • Interpret uniform convergence of functions and Stone-Weierstrass theorem
  • Evaluate the use of special functions
  • Elaborate the Contraction Mapping Theorem, The Inverse Function and the Implicit Function Theorems.


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