Definition:Hilbert Space

Definition
Let $V$ be an inner product space over $\Bbb F \in \left\{{\R, \C}\right\}$.

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 Spaces.

Standard Notation
In most of the literature, when studying a Hilbert space $H$, it is understood that:


 * $\left\langle{\cdot, \cdot}\right\rangle$ denotes the inner product on $H$;
 * $\left\|{\cdot}\right\|$ denotes the inner product norm on $H$;

unless specified otherwise.

Make sure to understand the precise definition of (especially) the inner product.

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.