Definition:Hilbert Space

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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.


Sources

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