Hilbert space

Pre-Hilbert space

A vector space ${ X }$ over the field ${ F=\mathbb{R}, \mathbb{C} }$ equipped with

\[(\cdot,\cdot) : X \times X \to F\]

satisfying the following conditions:

${ (x+y,z) = (x,z) + (y,z), \quad (\lambda x,y) = \lambda(x,y) }$ for all ${ \lambda \in F, x,y,z \in X }$
${ (y,x) = \overline{(x,y)} }$ for all ${ x,y \in X }$
${ (x,x) > 0 }$ for all nonzero ${ x \in X }$

Remark ${ (x,x) = 0 \mbox{ iff } x=0 }$

Hilbert space is a complete inner product space, i.e., the norm ${ \lVert x \rVert := \sqrt{(x,x)} }$ induces a metric

\[d(x,y) = \lVert x-y \rVert\]

Then we say ${ X }$ is Hilbert space if complete with ${ d }$.