This module is an introduction to functional analysis on Hilbert spaces. The subject of functional analysis builds on the linear algebra studied in the first year and the analysis studied in the second year.
The module introduces the concept of Hilbert spaces - an infinite dimensional analogue of Euclidean space. The definition of a vector space only allows the construction of finite sums of vectors. By introducing the norm of a vector (i.e. the length of a vector), it is then possible to do analysis with vectors, compute limits, take infinite sums etc.
In mathematics, the study of Hilbert spaces has applications including Fourier series, the Fourier transform, and the solution of differential equations. Hilbert spaces also play an important role in
physics, forming the mathematical basis for Quantum Mechanics
The module builds up to classical results such as the Toeplitz index theorem, the spectral theory of compact operators, and applications to differential operators.
Pre-requisites: (MATH1049 AND MATH2039 AND MATH2049) OR (MATH1049 AND MATH2039 AND MATH2046)