𝔯𝔢𝔞𝔩 𝔞𝔫𝔞𝔩𝔶𝔰𝔦𝔰

The problems below appeared on qualifying examinations in the graduate mathematics program at the University of Hawaii at Manoa.

Click here to see a list of the topics covered.

Functions

• elementary topology of \(ℝ^n\)

• 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

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𝔢𝔵𝔞𝔪𝔦𝔫𝔞𝔱𝔦𝔬𝔫𝔰

• Exam 1 (1–9)
• Exam 2 (10–16)
• Exam 3 (17–26)
• Exam 4 (27–35)
• Exam 5 (36–42)
• Exam 6 (43–49)
• Exam 7 (50–56)
• Exam 8 (57–63)
• Exam 9 (64–70)
• Exam 10 (71–76)

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𝔞𝔭𝔭𝔢𝔫𝔡𝔦𝔵

• background
• definitions
• theorems
• symbols

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𝔟𝔞𝔠𝔨 𝔪𝔞𝔱𝔱𝔢𝔯

• References
• acknowledgements
• INDEX

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Complex Analysis Exams

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