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.

Comments

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