# Definition:Hilbert Space

## Definition

Let $H$ be a vector space over $\mathbb F \in \set {\R, \C}$.

### Definition 1

Let $\struct { H, \innerprod \cdot \cdot_H }$ be an inner product space.

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

Let $\struct {H, d}$ be a complete metric space.

Then $H$ is a Hilbert space over $\mathbb F$.

### Definition 2

Let $\struct {H, \norm {\,\cdot\,}_H}$ be a Banach space with norm $\norm {\,\cdot\,}_H : H \to \R_{\ge 0}$.

Let $H$ have an inner product $\innerprod \cdot \cdot_H : H \times H \to \C$ such that the inner product norm is equivalent to the norm $\norm {\,\cdot\,}_H$.

Then $H$ is a Hilbert space over $\mathbb F$.

## Notation

The subscripts of the inner product $\innerprod \cdot \cdot_H$ and the inner product norm $\norm {\,\cdot\,}_H$ on $H$ serve to emphasize the space $H$ when considering multiple Hilbert spaces.

If it is clear from context which Hilbert space we are studying, the subscripts may be omitted such that::

$\innerprod \cdot \cdot$ or $\innerprod \cdot \cdot_H$ will denote the inner product on $H$
$\norm {\,\cdot\,}$ or $\norm {\,\cdot\,}_H$ will denote the inner product norm on $H$

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

## 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.