MAA 5229 Modern Analysis 2
The text for the course is Rudin, Principles of Mathematical Analysis (third edition). The topics covered fall into three modules, as follows.
- The Riemann (or Riemann-Stieltjes) integral: construction and elementary properties, including existence for monotonic functions and continuous functions. The relationship with the derivative, including the Fundamental Theorem(s) of Calculus.
- Sequences and series of functions. Uniform convergence; its interaction with continuity, integration and differentiation. Equicontinuous families and the Arzela-Ascoli theorem. The Stone-Weierstrass theorem and applications. Power series; examples, including the exponential and trigonometric functions.
- Lebesgue theory. Algebras and sigma-algebras of sets. Additive set functions and measures; examples, including Lebesgue measure. Measurable functions. The Lebesgue integral: its construction and elementary properties. The Monotone Convergence theorem; the Dominated Convergence theorem; the Fatou lemma. Square-integrability and Fourier series.
Note: These modules roughly correspond to chapters of Rudin, thus: (1) Chapter 6; (2) Chapters 7 and 8; (3) Chapter 11. Students are encouraged to supplement their study of the text by attempting problems from previous First-year examinations (for which, see http://gma.math.ufl.edu/); selections from the past five years or so ought to suffice (though one or two past examinations should be left unseen, for use as practice).