MAT3082

Hilbert spaces

Coordinator: Dr Robert Griffiths

4 points
* Two 1-hour lectures per week
* Second semester
* Clayton
* Prerequisites: MAT2020, MAT2051
* Prohibitions: MAP3082

Objectives On the completion of this subject students will be able to understand the basic concepts of Banach and Hilbert spaces; in particular recognise orthonormal systems and apply the Gram-Schmidt orthonormalisation process; master some basic properties of complete orthonormal systems and Fourier expansions and their relation to classical (trigonometric) Fourier expansions; understand the concepts and basic properties of bounded linear operators and functionals, including the Riesz representation theorem and the adjoint of an operator; know elements of spectral theory, in particular for self-adjoint operators.

Synopsis Inner product spaces, orthogonality. Hilbert spaces. Orthonormal systems, Gram-Schmidt process, Fourier expansions. Bounded linear operators and functionals. Riesz representation theorem. Spectral theory for self-adjoint operators: applications, for example to quantum mechanics and differential equations.

Assessment Examination (2 hours): 70%
* Assignments: 30%

Recommended texts

Young N An introduction to Hilbert spaces CUP, 1988