Module manager: Vladimir V. Kisil
Email: V.Kisil@leeds.ac.uk
Taught: Semester 1 (Sep to Jan) View Timetable
Year running 2026/27
MATH1000 Core Mathematics, and MATH1110 Real Analysis or equivalent
| MATH1000 | Core Mathematics |
| MATH1110 | Real Analysis |
This module is not approved as an Elective
Solving problems in infinite dimensional space is fundamental to our understanding of the world; finding the optimal depth for a wine cellar, or the natural frequencies of an object, fall within the same mathematical playground of Functional Analysis. Many such problems admit solutions in the form of an infinite sum or integral expression, but how(?), why(?), and are these expressions genuine? We will develop the mathematical theory to rigorously ask and answer these fundamental questions.
The module aims include consolidating the fundamental mathematical concepts of linear analysis in infinite‑dimensional vector spaces; developing a geometrized view of analysis based on magnitude and orthogonality derived from inner products in 2D and 3D Euclidean spaces; expressing boundary and initial value problems for selected partial differential equations within an abstract framework of linear function spaces; and give students practical skills for solving and analysing problems from a rigorous, pure‑mathematical standpoint.
Subject specific learning outcomes:
On successful completion of the module students will have demonstrated the following learning outcomes relevant to the subject:
Calculate the Fourier coefficients of certain elementary functions
Compute the Fourier transform of elementary functions on R
Perform a range of calculations involving orthogonal expansions in Hilbert spaces and Fourier integrals
Apply functional analytic techniques to the study of Fourier series and Fourier integrals.
Give the definitions and basic properties of various classes of operators (including the classes of compact, self-adjoint, and unitary operators) on a Hilbert space, and use them in specific examples
Prove results related to the theorems in the course.
Skills learning outcomes:
On successful completion of the module students will have demonstrated the following skills learning outcomes:
Accurately utilise relevant definitions to identify examples and construct non-examples of objects appearing in functional analysis.
Write proofs of statements relating to functional analysis and its applications.
Recall appropriate fundamental results covered in the course and apply them to solve equations and mathematical problems.
Appraise and explain the impact of the main theorems to the wider mathematical community
Real and complex Fourier series: the vibrating string example.
Banach and Hilbert spaces: subspaces, linear spaces and orthogonal complements.
Applications to Fourier series, the heat equation and the wave equation.
Linear operators and their spectra: operator norms, dual spaces, the adjoint (Hermitian, normal and unitary operators) and the spectral radius formula.
Compact and Hilbert-Schmidt operators.
Recap of measure theory and integration. Lp spaces and their duals.
Additional topics that build on these may be covered as time allows.
The Fourier transform and its mapping properties.
The spectral theorem for compact self-adjoint operators.
Topics from advanced PDE theory.
| Delivery type | Number | Length hours | Student hours |
|---|---|---|---|
| Lecture | 33 | 1 | 33 |
| Private study hours | 117 | ||
| Total Contact hours | 33 | ||
| Total hours (100hr per 10 credits) | 150 | ||
117
Formative feedback will be provided on regular example sets or other similar learning activity.
Check the module area in Minerva for your reading list
Last updated: 12/05/2026
Errors, omissions, failed links etc should be notified to the Catalogue Team