Department of Mathematics

The text for the course is Rudin, Principles of Mathematical Analysis (third edition). The topics covered fall into three modules, as follows.

1. 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.
2. 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.
3. 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).