Real Analysis Exams

These exams were taken by graduate students in mathematics at the University of Hawaii at Manoa.

Appendices and Indices


Topics. Expand to see a list of some topics covered on the exams.

Functions

  • elementary topology of \(\mathbb R^n\)
  • continuous functions
  • uniformly continuous functions
  • absolutely continuous functions
  • differentiability, implicit function theorem
  • functions of bounded variation, rectifiable curves
  • Borel functions
  • measurable functions
  • simple functions
  • relations between these classes, Lusin’s theorem

Integration

  • Lebesgue integral
  • Fatou’s lemma
  • monotone and dominated convergence theorems
  • applications to moving limits through integrals
  • differentiation under an integral sign
  • product measures, Fubini-Tonelli theorem
  • definition and completeness of \(L^p\) spaces
  • Riesz representation theorem
  • Lebesgue differentiation theorem

Convergence of functions

  • pointwise convergence
  • supremum norm, uniform convergence
  • convergence in measure
  • convergence in \(L^p\) spaces
  • relations between these notions, Egorov’s theorem

Inequalities

  • Chebyshev
  • Cauchy-Schwarz
  • Jensen
  • Hölder
  • Minkowski (sum and integral forms)

Density

  • Weierstrass theorem, density of polynomials in \(L^p\) spaces
  • convolution with approximate identities
  • density of smooth functions in \(L^p\) spaces

Measure theory

  • Borel σ-algebra, general σ-algebras
  • construction of Lebesgue measure on Euclidean spaces
  • outer measures, counting measure, product measures
  • \(L^p\) spaces of a general measure space
  • absolute continuity of measures
  • Radon–Nikodym theorem


Other excellent sources.

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Complex Analysis Exams

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