# Examination topics in the 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.