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# Functional Analysis - MAT00107M

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• Department: Mathematics
• Credit value: 20 credits
• Credit level: M
• Academic year of delivery: 2024-25
• See module specification for other years: 2023-24

## Module summary

An introduction to Hilbert Space and the properties of bounded and compact linear maps between Hilbert Spaces.

## Related modules

• None

### Prohibited combinations

• None

MSc students will need to be, or become by extra reading, familiar with:

From Linear Algebra:
Vector spaces over the real and complex numbers, subspaces, linear (in)dependence, spanning sets, finite bases, finite dimension, inner products, linear maps, rank and nullity, matrices, (conjugate) transpose, eigenvalues and eigenvectors.

From Metric Spaces:

Open and closed sets, limits of sequences and functions, continuity (including inverse images of open and closed sets), Cauchy sequences and completeness, compactness including Bolzano-Weierstrass property.

From Measure and Integration:

Measurability and integrability, null sets, basic properties of the integral (linearity etc.), Lebesgue measure in at least 2 dimensions, absolutely convergent sums thought of as integrals on a space of atoms, Monotone and Dominated Convergence Theorems, Fubini and Tonelli Theorems.

## Module will run

Occurrence Teaching period
A Semester 2 2024-25

## Module aims

An introduction to Hilbert Space and the properties of bounded and compact linear maps between Hilbert Spaces.

## Module learning outcomes

By the end of the module, students should be able to:

1. Work with orthogonal decompositions, projections and bases to establish results about abstract and specific Hilbert spaces.

2. Determine whether or not certain examples of linear operators defined on (subspaces of) Hilbert spaces are bounded or compact; find adjoints of bounded operators; obtain spectral information about specific operators or classes of operators.

3. Determine spectra of certain examples of linear operators, including the use of the functional calculus of self-adjoint bounded operators.

## Module content

• Hilbert spaces: definition and characterisation, orthogonal decompositions and projections, orthonormal bases.

• Bounded linear operators: definition and characterisation, self-duality of Hilbert space, adjoint operators, self-adjoint operators, unitary operators.

• Spectral theory: eigenvalues, eigenvectors, the spectrum, spectra of self-adjoint and unitary operators, spectral theorem for compact self-adjoint operators, spectral mapping, functional calculus.

• Some applications to differential equations.

## Indicative assessment

Task % of module mark
Closed/in-person Exam (Centrally scheduled) 100

None

### Indicative reassessment

Task % of module mark
Closed/in-person Exam (Centrally scheduled) 100

## Module feedback

Current Department policy on feedback is available in the student handbook. Coursework and examinations will be marked and returned in accordance with this policy.

Nicholas Young “An introduction to Hilbert Space” CUP 1988

The information on this page is indicative of the module that is currently on offer. The University constantly explores ways to enhance and improve its degree programmes and therefore reserves the right to make variations to the content and method of delivery of modules, and to discontinue modules, if such action is reasonably considered to be necessary. In some instances it may be appropriate for the University to notify and consult with affected students about module changes in accordance with the University's policy on the Approval of Modifications to Existing Taught Programmes of Study.