Syllabus for the Comprehensive Exam in Analysis

  1. Measure Spaces: sigma-algebras, including the sigma-algebras of Borel and Lebesgue measurable sets; measures, including the counting measure and Lebesgue measure; finite and sigma-finite measures; signed measures, complex measures, and product measures
  2. Integration Theory: The Riemann integral; the Lebesgue integral; integration with respect to a measure or signed measure; modes of convergence: in measure, almost surely, and in L^p; convergence theorems: Fatou's lemma, the monotone convergence theorem, and Lebesgue's dominated convergence theorem; product measures and the theorems of Fubini and Tonelli; absolutely continuous measures and the Radon-Niko-dym theorem; singular measures and the Lebesgue decomposition
  3. Function Theory: Monotone functions and functions of bounded variation; differentiation, absolutely continuous functions, and the fundamental theorem of calculus
  4. Topological Spaces: The real number system; metric spaces, including completeness, abstract topological spaces, and the Baire category theorem and its consequences; compact spaces and the Tychonoff theorem
  5. The Classical Function Spaces: L^p and l^p spaces for 1 ≤ p ≤ ∞; C(K) spaces; Hölder's and Minkowski's inequalities; bounded linear functionals on L^p and l^p for 1 ≤ p < ∞ and on C(K); Ascoli's theorem; the Stone-Weierstrass theorem
  6. Elementary Functional Analysis: Hilbert spaces; the projection theorem; applications to approximation; completeness and orthonormal bases; representation of bounded linear functionals on Hilbert space; Banach spaces; the Hahn-Banach, closed graph, and open mapping theorems


Suggested textbook: Real Analysis by Folland
Suggested courses: 6337 and 6338
Other relevant courses: 4317, 4318, and 7338