# 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

$$\| x+ y \| ^ {2} + \| x \cdot y \| = \ 2 ( \| x \| ^ {2} + \| y \| ^ {2} ) .$$

For the characterizations of inner product spaces see [a1], Chapt. 4.

#### References

 [a1] V.I. Istrăţescu, "Inner product structures" , Reidel (1987) [a2] W. Rudin, "Functional analysis" , McGraw-Hill (1979) [a3] K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, Sect. 4; 5
How to Cite This Entry:
Pre-Hilbert space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pre-Hilbert_space&oldid=48275
This article was adapted from an original article by V.I. Lomonosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article