Real Analysis Qualifying Examination Syllabus
It is expected that the student has mastered the basic principles of analysis of real functions of one and several variables. It includes, among other topics, knowledge of the Riemann integral, the differential, the Jacobian matrix, the implicit function theorem, and Stokes' theorem. In addition, the following outlined topics from the theory of functions of a real variable should be mastered.
- Basic properties of measurable sets and functions.
- measurable sets and functions,
- Lebesgue measure and Lebesgue integral,
- Lebesgue-Stieltjes integral.
- Convergence properties of measurable functions and integration.
- algebraic and convergence properties of measurable functions,
- convergence theorems for integrals: Fatou's Lemma, monotone convergence theorem, dominated convergence theorem,
- measure and outer measure including product measures and Fubini's theorem.
- functions of bounded variation,
- absolutely continuous function and indefinite integrals,
- singular functions,
- Lebesgue decomposition,
- Radon-Nikodym derivative.
- Banach and Hilbert space
- Holder and Minkowski inequalities,
- L-p spaces,
- representation of bounded linear functions on Hilbert and L-p spaces,
- orthonormal families and the Riesz-Fischer theorem,
- complete orthonormal systems and Fourier series,
- linear functions and the Hahn-Banach theorem, Baire's theorem and its consequences: the Banach-Steinhaus, open mapping and closed graph theorem.
- Theory of Differentiation. This includes:
- Classical Banach and Hilbert Space Theory. This includes:
- W. Rudin. Principles of Mathematical Analysis.
- W. Rudin. Real and Complex Analysis. Chapters 1-6 (from Chapter 2 omit the Riesz representation theorem and from Chapter 5 an abstract approach to the Poisson Integral), Chapter 8 (omit convolutions and distribution functions).
- H. Royden. Real Analysis. Chapters 2-6 (omit Chapter 4, Section 5), Chapter 10 (omit Sections 5,6,7), Chapter 11 (omit Section 4), Chapter 12 (omit Sections 5,6,7,8,9).