MAA 6616–6617 Analysis Qualifying Exam Syllabus

The textbook is Sheldon Axler, Measure, Integration & Real Analysis.

Assumed background knowledge

Students are expected to have proficiency in metric-space topology, including continuity, compactness, and connectedness; the Riemann integral; and abstract linear algebra.

Fall semester (MAA 6616)

1. Measures: Lebesgue measure on the real line; measurable spaces and functions; convergence properties of measurable functions.
2. Integration with respect to a general measure and limit theorems; for example, monotone and dominated convergence and Fatou’s lemma.
3. Lebesgue differentiation and the Hardy–Littlewood maximal function.
4. Product measures, the iterated integral theorems, and integration in real Euclidean space.

Note: The Radon–Nikodym theorem is included in the spring semester syllabus.

Spring semester (MAA 6617)

1. Banach spaces, including the Hahn–Banach theorem and consequences of the Baire category theorem.
2. Hilbert spaces.
3. L^p spaces. Hölder and Minkowski inequalities. Completeness.
4. Signed measures. Total variation; the Hahn, Jordan, and Lebesgue decomposition theorems; the Radon–Nikodym theorem; L^p–L^q duality.
5. Linear maps on Hilbert spaces. Adjoints, the spectrum, and the spectral theorem for compact operators.