The process of extending the algebraic and geometrical methods of linear algebra from matrices to differential or integral operators consists of going from a finite dimensional vector space, typically , to an infinite dimensional vector space, typically a function space.
However, a vector space of functions has certain idiosyncrasies precisely because its dimension is infinite. These peculiarities are so important that we must develop the framework in which they arise.
One of the most useful, if not the most useful, framework is the theory of Hilbert spaces, the closest thing to the familiar finite-dimensional Euclidean spaces. In passing we shall also mention metric spaces and Banach spaces.
We shall see that infinite dimensional vector spaces are a powerful way of organizing the statement and solution of boundary value problems. In fact, these spaces are the tool of choice whenever the linear superposition principle is in control. This happens in signal processing, in quantum mechanics, electromagnetic wave theory, and elsewhere.
The most notable peculiarity associated with infinite dimensional vector spaces is the issue of completeness.
From the viewpoint of physics and engineering, completeness is an issue of accuracy. We would like to have at our disposal mathematical concepts which are such that they are capable of expressing the natural phenomenon (under consideration) no matter how accurately it is described now or will be described any time in the future.
One of the most useful infinite dimensional vector spaces is Hilbert space. To define it, we must have at our disposal the constellation of concepts on which it is based. Let us identify the components of the constellation.