Functional Analysis - MAT00045M

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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.

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 (S 7.82 YOU).

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