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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.
Any failed components are reassessed, if necessary, by an exam in the reassessment 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, continuous maps, complete metric spaces.
• Topological spaces: Hausdorff, connected and compact
spaces.
• Normed vector spaces: operator norms and norms on
finite-dimensional vector spaces.
• Infinite series: 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.