MATH 6110 Real Analysis

Main Topics

1. Measures and abstract integration.
2. Lebesgue measure, L^p spaces.
3. Hilbert spaces, Banach spaces.
4. Fourier series. (Optional: Fourier transforms, Fourier inversion, and Plancherel theorems)
5. Differentiation.
6. Integration on product spaces, Fubini’s theorem.

Optional topics

1. Introduction to probability — probabilistic terminology, Borel-Cantelli, strong law of large numbers (1-2), independence (8), central limit theorem (9), conditional expectation (6).  This could be interwoven with the syllabus above. The numbers refer to where these ideas fit into the syllabus above.
2. More functional analysis.
3. Generalized functions.