Aims

To introduce to students the similarities and differences between linear operators in finitely many and infinitely many dimensions.

Learning objectives

• 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.

Syllabus

• The algebra of bounded operators on a Hilbert space and the ideals of finite-rank, Hilbert-Schmidt and compact operators.
• Definitions and properties of self-adjoint and unitary operators.
• The spectral theorem and functional calculus. Exact details may vary from year to year, but this will normally cover either compact self-adjoint operators or bounded normal operators.

Recommended texts

• * 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).

Teaching

• Two lectures per week
• One problem class per week

Assessment

Two hour closed examination in week 1 of the Spring Term (90%)
Coursework (10%). Note that coursework submitted after the advertised deadlines will be given a mark of zero.

Elective information

This module is not available as an elective.

Prerequisites

• Topics in Analysis
• Linear Algebra