ProofWiki:Sandbox/Definition:Hilbert Space

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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.