The list of topics for the entry exam to the AGH Doctoral School in discipline Mathematics

  • Closure of the spectrum of a continuous operator in a Banach space.
  • Uniform Boundedness Principle (key points of the proof).
  • Fourier series decomposition for square integrable functions (types of convergence).
  • Operator norm convergence and strong convergence of a sequence of operators.
  • Relationship between symmetric and self-adjoint operators.
  • Properties of the spectrum of compact operators.
  • Example of a spectral measure, spectral theorem.
  • Examples of applications of the Hahn-Banach theorem.
  • Definition of the measurable space, Borel σ-algebra.
  • Justification that continuous functions are measurable with respect to Borel σ-algebra.
  • Proof that every measurable function is the limit of a sequence of simple and measurable functions.
  • Definition of the measure. Continuity property of measure.
  • The construction of Lebesgue measure (in Rn).
  • Lebesgue integral of bounded function.
  • Lebesgue dominated convergence theorem.
  • Differentiation in the complex domain. The Cauchy-Riemann equations.
  • The Cauchy integral theorem.
  • Power series expansions of analytic functions.
  • Laurent series expansions of functions.
  • Uniqueness theorem for analytic functions.
  • Definition of topological space. Examples.
  • Definitions of continuous functions in topological spaces. Homeomorphisms.
  • Cartesian product of topological spaces. Tychonoff topology.
  • Definition of a net (Moore–Smith sequence). Convergence in topological spaces.
  • Justification that the continuous image of a compact set is compact.