# Pre-Hilbert space

A vector space $E$ over the field of complex or real numbers equipped with a scalar product $E \times E \rightarrow \mathbf C$, $x \times y \rightarrow ( x , y )$, satisfying the following conditions:

1) $( x + y , z ) = ( x , z ) + ( y , z )$, $( \lambda x , y ) = \lambda ( x , y )$, $( y , x ) = \overline{ {( x , y ) }}\;$, $x , y , z \in E$, $\lambda \in \mathbf C$( $\mathbf R$);

2) $( x , x ) \geq 0$ for $x \in E$;

3) $( x, x) = 0$ if and only if $x = 0$.

On a pre-Hilbert space a norm $\| x \| = ( x , x ) ^ {1/2}$ is defined. The completion of $E$ with respect to this norm is a Hilbert space.

A function $( x, y)$ as above is also called an inner product. If it satisfies only 1) and 2) it is sometimes called a pre-inner product. Accordingly, pre-Hilbert spaces are sometimes called inner product spaces, while vector spaces with a pre-inner product are also called pre-inner product spaces.
If $( E, \| \cdot \| )$ is a normed linear space, then it has an inner product generating the norm if (and only if) the norm satisfies the parallelogram law
$$\| x+ y \| ^ {2} + \| x \cdot y \| = \ 2 ( \| x \| ^ {2} + \| y \| ^ {2} ) .$$