Definition:Hilbert Space

Definition
Let $V$ be an inner product space over a subfield $\Bbb F$ of $\C$.

Let $d: V \times V \to \R_{\ge 0}$ be the metric induced by the inner product norm.

If $\left({V, d}\right)$ is a complete metric space, $V$ is said to be a Hilbert space.

Results about Hilbert spaces are categorised in Category:Hilbert Space Theory.

Historical Note
Hilbert spaces were among the first attempts to generalise the Euclidean spaces $\R^n$.

Study of these objects eventually led to the development of the field of functional analysis.