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Hilbert Spaces - MAT00063M

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• Department: Mathematics
• Credit value: 10 credits
• Credit level: M
• Academic year of delivery: 2022-23

Related modules

Additional information

Pre-requisite knowledge for MSc students: basic knowledge of linear algebra, metric spaces and Lebesgue integration.

Module will run

Occurrence	Teaching period
A	Autumn Term 2022-23

Module aims

A Hilbert Space is an inner product space which is complete, in the sense that every Cauchy sequence converges. Hilbert spaces are an important tool in the study of Fourier series and transforms, integral and differential equations, and quantum theory. One of the main aims of this module is to develop the idea of a Fourier series in the context of a Hilbert space, and to show that classical Fourier series, introduced in first-year Calculus, fit into this general framework. To do this, we introduce an important example of a Hilbert space (known as L²) which is constructed using the Lebesgue integral.

Module learning outcomes

The study of Hilbert spaces introduces students to some advanced techniques in analysis and serves as a good introduction to the sort of thinking required for research. As well as this, Hilbert spaces are important in many other areas of mathematics, including the studying of Quantum Mechanics, so a successful student will be able to apply their techniques and knowledge across disciplinary borders.

Cognitive & Intellectual Skills: Analysis, Synthesis, Evaluation, Application, all developed through learning new techniques and applying them to complex problems.

Key/Transferable Skills: Management of information – this module will equip students for research in analysis; Autonomy and Problem Solving developed through regular coursework assignments and tested in the examination.

Technical Expertise: the skills and techniques developed in this module are widely applicable across mathematics (analysis, mathematical physics).

Module content

Topics covered:

• The definition of a Hilbert space, as compared to an incomplete inner product space, and examples of both.
• The use of orthogonal decompositions and projections in Hilbert spaces.
• The idea of an orthonormal sequence in a Hilbert space and the various equivalent formulations of completeness of such a sequence.
• The use of complete orthonormal sequences to show that any two separable infinite-dimensional Hilbert spaces are isomorphic.
• Some important examples of complete orthonormal sequences, especially the exponential system and its relationship with classical Fourier series.

 

Indicative assessment

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

Special assessment rules

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.

Indicative reading

N Young, An Introduction to Hilbert Space, Cambridge University Press.

A J Weir, Lebesgue Integration and Measure, Cambridge University Press.