# 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
• 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