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College of Liberal Arts and Sciences University of Florida

MAA 5228 Modern Analysis 1

The text for the course is Walter Rudin, Principles of Mathematical Analysis, 3rd edition. The topics covered fall into five modules, as follows.

  1. A review of the real number system, including suprema and infima; a review of the complex number system, including the Cauchy–Schwarz inequality.
  2. The notions of countability and uncountability. Metric spaces; open sets and closed sets. Compactness; the Heine–Borel and Bolzano–Weierstrass theorems. Connectedness.
  3. Sequences in metric spaces; subsequences and convergence. Cauchy sequences and complete spaces. Real sequences, including limsup and liminf, with examples. Real and complex series: convergence and absolute convergence.
  4. Continuous functions between metric spaces. Uniform continuity and continuity on compact spaces. Continuity on connected spaces and the intermediate value theorem. Discontinuities and monotonic functions.
  5. Differentiation of real-valued functions. Fundamental properties, including the chain rule. The mean value theorem, with applications including l’Hôpital’s rule. Higher derivatives and Taylor’s theorem.

Note: These modules (in order) closely match the first five chapters of Rudin. Students are encouraged to supplement their study of the text by attempting problems from previous first-year analysis examinations; 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).