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Functional Analysis - MAT00045M

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  • Department: Mathematics
  • Module co-ordinator: Prof. Chris Fewster
  • Credit value: 10 credits
  • Credit level: M
  • Academic year of delivery: 2022-23
    • See module specification for other years: 2021-22

Related modules

Pre-requisite modules

Co-requisite modules

  • None

Prohibited combinations

  • None

Additional information

Summary of prerequisite topics for MSc students: in addition to the Autumn-term Hilbert Space module: basics of Linear Algebra including unitary diagonalization; ideas from metric or normed spaces including continuity, completeness and compactness, orthogonal decompositions and orthogonal bases of Hilbert spaces.

Module will run

Occurrence Teaching period
A Spring Term 2022-23

Module aims

  • To explore the exotic world of linear operators in infinitely many dimensions.

  • To compare linear operators in finitely many and infinitely many dimensions.

  • To encounter specific applications of the general theory.

Module learning outcomes

  • The definition and significance of bounded and compact operators.

  • The idea of the spectrum, and the difference between 'spectral point' and 'eigenvalue'.

  • The Spectral Theorem and functional calculus as a generalisation of the orthogonal diagonalisation theorem.

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Module content

 

Syllabus

  • The algebra of bounded operators in a Hilbert space and the ideals of finite-rank, Hilbert-Schmidt and compact operators.

  • Definitions and properties of self-adjoint operators.

  • Spectral Theorem and functional calculus illustrated by compact normal operators or bounded self-adjoint operators.

  • Applications including differential or integral equations.

Assessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Functional Analysis		 2 hours 100

Special assessment rules

None

Reassessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Functional Analysis		 2 hours 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 (S 7.82 YOU).

E Kreysig, Introductory Functional Analysis with Applications, Wiley (S 7.8 KRE).


The information on this page is indicative of the module that is currently on offer. The University is constantly exploring 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 by the University. Where appropriate, the University will notify and consult with affected students in advance about any changes that are required in line with the University's policy on the Approval of Modifications to Existing Taught Programmes of Study.