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𝔯𝔢𝔞𝔩 𝔞𝔫𝔞𝔩𝔶𝔰𝔦𝔰
================

The problems below appeared on qualifying examinations in the graduate mathematics program at the `University of Hawaii at Manoa <http://math.hawaii.edu/wordpress/graduate/>`__.

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  .. container:: header

     Click here to see a list of the topics covered.

  **Functions**

  + elementary topology of :math:`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 :math:`L_p` spaces
  + Riesz representation theorem
  + Lebesgue differentiation theorem

  **Convergence of functions**

  + pointwise convergence
  + supremum norm, uniform convergence
  + convergence in measure
  + convergence in :math:`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 :math:`L_p` spaces
  + convolution with approximate identities
  + density of smooth functions in :math:`L_p` spaces

  **Measure theory**

  + Borel σ-algebra, general σ-algebras
  + construction of Lebesgue measure on Euclidean spaces
  + outer measures, counting measure, product measures
  + :math:`L_p` spaces of a general measure space
  + absolute continuity of measures
  + Radon–Nikodym theorem

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𝔢𝔵𝔞𝔪𝔦𝔫𝔞𝔱𝔦𝔬𝔫𝔰
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.. toctree::
    :maxdepth: 1

    1991Nov21.rst
    1994Nov16.rst
    1995Nov17.rst
    1998Apr03.rst
    1998Jun23.rst
    2000Nov17.rst
    2001Nov26.rst
    2004Apr19.rst
    2006Apr.rst
    2007Nov16.rst

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𝔞𝔭𝔭𝔢𝔫𝔡𝔦𝔵
--------

.. toctree::
    :maxdepth: 1

    appendix_background
    appendix_definitions
    appendix_theorems
    appendix_symbols

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𝔟𝔞𝔠𝔨 𝔪𝔞𝔱𝔱𝔢𝔯
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.. toctree::
    :maxdepth: 1

    refs
    acknowledgement
    genindex


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.. * :ref:`modindex`
.. * :ref:`search`