On successful completion of this module, students will be able to:
Accurately recall definitions, state theorems and produce proofs on topics in metric spaces normed vector spaces and topological spaces;
Construct rigorous mathematical arguments using appropriate concepts and terminology from the module, including open, closed and bounded sets, convergence, continuity, norm equivalence, operator norms, completeness, compactness and connectedness;
Solve problems by identifying and interpreting appropriate concepts and results from the module in specific examples involving metric, topological and /or normed vector spaces;
Construct examples and counterexamples related to concepts from the module which illustrate the validity of some prescribed combination of properties;
Discuss countable sets, characteristic functions and bolean algebras;
State and prove properties of length measure, outer measure and Lebesgue measure for subsets of the real line and establish measurability for a range of functions and sets;
Define the Lebesgue integral on the real line and apply basic results including convergence theorems.
Module Content
Metric spaces (including open and closed sets, continuous maps and complete metric spaces);
Normed vector spaces (including operator norms and norms on finite dimensional vector spaces);
Topological properties of metric spaces (including Hausdorff, connected and compact spaces);
Countable versus uncountable sets; inverse images;characteristic functions; boolean algebra for subsets.
Algebras of subsets of the real line; length measure on the interval algebra; finite-additivity; subadditivity and countable-additivity; outer measure; Lebesgue measurable sets; extension to sigma algebra; Borel sigma algebra.
Lebesgue measurable functions; simple functions; integrals for non-negative functions; limits of measurable functions and the monotone convergence therorem; Lebesgue integrable functions; generalisation of the Riemann integral (for continuous functions on finite closed intervals).
Fatou's lemma; dominated convergence theorem; integrals depending on a parameter
Module Prerequisite
Recommended Reading
Introduction to metric and topological spaces, W.A. Sutherland. Oxford University Press, 1975;
Metric Spaces, E.T. Copson. Cambridge University Press, 1968;
Assessment
This module will be examined in a 3-hour examination. Continuous assessment in the form of homework assignments will contribute 15% to the final grade at the annual examinations, with the examination counting for the remaining 85%. Supplementals if required will consist of 100% exam.