# School of Mathematics

## Language

Credit weighting (ECTS)
10 credits
Semester/term taught
Michaelmas & Hilary Term 2019-20
Contact Hours

Lecturer
Prof. John Stalker  Prof. David Wilkins
Learning Outcomes
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