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MAU22200 Advanced analysis

Module Code	MAU22200
Module Title	Advanced analysis
Semester taught	Semesters 1,2 (yearlong)
ECTS Credits	10
Module Lecturer	Prof. Katrin Wendland
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 15% towards the overall mark.
The module is passed if the overall mark for the module is 40% or more. If the overall mark for the module is less than 40% and there is no possibility of compensation, the module will be reassessed as follows:
1) A failed exam in combination with passed continuous assessment will be reassessed by an exam in the supplemental session;
2) The combination of a failed exam and failed continuous assessment is reassessed by the supplemental exam;
3) A failed continuous assessment in combination with a passed exam will be reassessed by one or more summer assignments in advance of the supplemental session.

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, convergent sequences, uniform convergence, continuous maps, complete metric spaces, Banach fixed point theorem.
• Topological spaces: Hausdorff, connected and compact spaces.
• Normed vector spaces: bounded operators, operator norms and norms on finite-dimensional vector spaces, Banach spaces.
• Infinite series: absolute versus conditional convergence, countable versus uncountable sets, double series, convergence criteria.
• Contents and measures: Boolean algebra for subsets, sigma algebra, Borel sigma algebra, content space, outer measure, measure, Lebesgue measurable sets.
• Lebesgue integral: Lebesgue measurable functions, simple functions, Lebesgue integrable functions, limits of measurable functions, monotone and dominated convergence theorems, Fatou's Lemma, Cavalieri's Principle, Fubini-Tonelli Theorem.