MAA 6616-7 Analysis

Suggested references

There is no prescribed text, but the following textbooks are suggested:

  • Gerald B. Folland, Real Analysis: Modern Techniques and their Applications, Wiley, 1999
  • Walter Rudin, Real and Complex Analysis, McGraw-Hill, 1986

Topics

The PhD examination covers both abstract integration theory and basic
functional analysis. The precise content varies with instructor, but
will include the following:

  • Measure: σ-algebras, construction of measures (the Caratheodory
    and Hahn-Kolmogorov theorems), Lebesgue measure on \(\mathbb{R}^n\), product
    measures, signed measures and the Hahn-Jordan decomposition
  • Integration: measurable functions, the Lebesgue integral, modes of
    convergence, the Vitali convergence theorem, the Fubini-Tonelli
    theorem, the Radon-Nikodym theorem
  • Functional Analysis: normed vector spaces and Banach spaces, bounded
    linear operators, the Hahn-Banach theorem and its corollaries, dual
    spaces, the Baire Category theorem and its applications (Open Mapping
    theorem, Closed Graph theorem, Banach isomorphism theorem), the
    Banach-Steinhaus theorem, Hilbert spaces, \(L^p\) spaces.

As noted above, additional topics may appear, according to time and
taste; these have included the Lebesgue Differentiation Theorem,
Banach-Mazur distance between normed spaces, and basic results in
Fourier analysis and probability. Students preparing to take the
examination should consult the most recent instructor of the course
for additional information.

(posted April 9, 2014)