Functional Analysis - MAT00045M
- Department: Mathematics
- Credit value: 10 credits
- Credit level: M
- Academic year of delivery: 2022-23
Related modules
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 |
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A | Spring Term 2022-23 |
Module aims
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To explore the exotic world of linear operators in infinitely many dimensions.
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To compare linear operators in finitely many and infinitely many dimensions.
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To encounter specific applications of the general theory.
Module learning outcomes
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The definition and significance of bounded and compact operators.
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The idea of the spectrum, and the difference between 'spectral point' and 'eigenvalue'.
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The Spectral Theorem and functional calculus as a generalisation of the orthogonal diagonalisation theorem.
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Module content
Syllabus
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The algebra of bounded operators in a Hilbert space and the ideals of finite-rank, Hilbert-Schmidt and compact operators.
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Definitions and properties of self-adjoint operators.
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Spectral Theorem and functional calculus illustrated by compact normal operators or bounded self-adjoint operators.
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Applications including differential or integral equations.
Indicative assessment
Task | % of module mark |
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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 (S 7.82 YOU).
E Kreysig, Introductory Functional Analysis with Applications, Wiley (S 7.8 KRE).