Esbjörn Runesson: Introduction to Hilbert spaces

Abstract.

This thesis revolves around the concept of Hilbert spaces which is an essential component within physics and mathematics relating to fields such as quantum mechanics, functional analysis, ergodic theory and probability theory. The focus of this thesis project is to give an introduction to the mathematical background which is needed to define Hilbert spaces. The first two chapters introduces the mathematical framework for the two main areas of this thesis project, namely metric topology and linear algebra. After establishing these mathematical ingredients we will move on to the last chapter where introduce terms such as a Hilbert base defined as a maximal orthonormal set in a Hilbert space, Hilbert dimension defined as the cardinality of the Hilbert base, where there exists one Hilbert space for every cardinality of a Hilbert base, and Fourier expansion which is indispensable when it comes to approximations of vectors within Hilbert spaces. The final subchapter is on the Riesz representation theorem which exemplifies the usefulness of Hilbert spaces.