# ProofWiki:Sandbox/Definition:Hilbert Space

## Definition

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

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

If $\struct {V, d}$ is a complete metric space, $V$ is said to be a Hilbert space.

## Standard Notation

In most of the literature, when studying a Hilbert space $\HH$, unless specified otherwise, it is understood that:

$\innerprod \cdot \cdot$ or $\innerprod \cdot \cdot_\HH$ denotes the inner product on $\HH$
$\norm {\,\cdot\,}$ or $\norm {\,\cdot\,}_\HH$ denotes the inner product norm on $\HH$

where the subscripts serve to emphasize the space $\HH$ when considering multiple Hilbert spaces.

## Also see

• Results about Hilbert spaces can be found here.

## Source of Name

This entry was named for David Hilbert.

## Historical Note

The Hilbert space was one of 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.