Module manager: Kasia Wyczesany
Email: k.b.wyczesany@leeds.ac.uk
Taught: Semester 1 (Sep to Jan) View Timetable
Year running 2026/27
MATH2150 Calculus, Curves and Complex Analysis
| MATH2150 | Calculus, Curves and Complex Analysis |
MATH3211 Metric and Function Spaces
This module is not approved as a discovery module
If you would like to undertake a rigorous study of a physical, geometrical, or statistical law, you will likely need both metrics and measures. Metric spaces are equipped with a generalised notion of distance between points, which is essential in modern analysis, while measure theory generalises familiar ideas of volume and underpins modern integration. In this module, we’ll explore these beautiful topics and prove results fundamental to pure and applied mathematics, including Picard–Lindelöf’s theorem from ODEs, the inverse and implicit function theorems, and Lebesgue’s dominated convergence theorem.
This module introduces students to the abstract study of spaces equipped with a notion of distance and the rigorous foundations of integration. Students will investigate the structure and properties of metric spaces—such as convergence, continuity, and compactness—and examine how these ideas support key analytical results, including Picard–Lindelöf’s theorem. They will also study the foundations of measure theory, exploring measurability, measurable functions, and integration, culminating in major results like Lebesgue’s dominated convergence theorem.
On successful completion of the module students will have demonstrated the following learning outcomes: 1. Accurately identify examples and non-examples of metric spaces and measure spaces. 2. Write proofs of statements relating to metric and measure spaces. 3. Recall appropriate fundamental results covered in the course and apply them to solve equations and mathematical problems. 4. Appraise and explain the impact of the main theorems to the wider mathematical community.
1. Metric spaces: definition and examples, inducing metrics from norms and inner products 2. Sequences in metric spaces: convergence, Cauchy sequences 3. Topology of metric spaces: open and closed sets, compactness and completeness 4. Continuous functions on metric spaces 5. Banach fixed-point theorem and application to ODEs 6. Measure Spaces 7. The Lebesgue measure and its properties 8. Measurable and non-measurable functions, properties of Lp spaces Additional topics that build on these may be covered as time allows. Such topics may be drawn from the following, or similar: · Further topics on normed spaces and dual spaces · Fourier transform · Banach spaces and linear functionals · Banach-Mazur distance: distance between lp and lq
| Delivery type | Number | Length hours | Student hours |
|---|---|---|---|
| Lecture | 44 | 1 | 44 |
| Private study hours | 156 | ||
| Total Contact hours | 44 | ||
| Total hours (100hr per 10 credits) | 200 | ||
There will be homework, with some problems marked for feedback only.
Check the module area in Minerva for your reading list
Last updated: 12/05/2026
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