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Module MAU22200: Advanced Analysis

Credit weighting (ECTS)
10 credits
Semester/term taught
Michaelmas & Hilary Term 2021-22
Contact Hours
 
Lecturer
Prof. John Stalker  
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 Jordan content, outer measure and Lebesgue measure for subsets of the Euclidean space and establish measurability for a range of functions and sets;
  • Define the Lebesgue integral on the Euclidean space 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 Euclidean space; Jordan content on the Jordan 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 theorem; Lebesgue integrable functions;
  • Fatou's lemma; dominated convergence theorem.
  • The fundamental theorem of calculus for Lebesgue integrals.
  • The Fubini-Tonelli theorem.

Module Prerequisite
MAU11204
 
Recommended Reading
  • Introduction to metric and topological spaces, W.A. Sutherland. Oxford University Press, 1975;
  • An Introduction to Measure Theory, Terence Tao. American Mathematical Society, 2011;
  • 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 20% to the final grade at the annual examinations, with the examination counting for the remaining 80%. Re-assessments if required will consist of 100% exam.