# 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 metric induced by the inner product norm $\norm {\,\cdot\,}_V$.

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

The Hilbert space $V$ may be considered as one of the following:

The complete inner product space $\struct {V, \innerprod \cdot \cdot_V}$
The Banach space $\struct {V, \norm {\,\cdot\,}_V}$

The topological space $\struct {V, \tau_d}$ where $\tau_d$ is the topology induced by $d$
The vector space $\struct {V, +, \circ}_{\Bbb F}$

That is to say, all theorems and definitions for these types of spaces directly carry over to all Hilbert spaces.

## 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.

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

Furthermore, the parentheses around the argument of linear functionals and linear transformations on $\HH$ are often suppressed for brevity.

Make sure to understand which symbols denote scalars, operators and functionals, respectively.

## 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.