MAU22200 Advanced analysis
| Module Code | MAU22200 |
|---|---|
| Module Title | Advanced analysis |
| Semester taught | Semesters 1,2 (yearlong) |
| ECTS Credits | 10 |
| Module Lecturer | Prof. John Stalker |
| Module Prerequisites | MAU11204 Analysis on the real line |
Assessment Details
- This module is examined in a 3-hour examination at the end of Semester 2.
- Continuous assessment contributes 20% towards the overall mark.
- Re-assessment, if needed, consists of 100% exam.
Contact Hours
11+11 weeks of teaching with 3 lectures and 1 tutorial per week.
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 Boolean 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: open and closed sets, continuous maps, complete spaces.
- Topological spaces: Hausdorff, connected and compact spaces.
- Normed vector spaces: operator norms and norms on finite-dimensional vector 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, 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.
Recommended Reading
- Introduction to metric and topological spaces by W.A. Sutherland.
- An introduction to measure theory by T. Tao.
- Metric spaces by E.T. Copson.