Pre-Hilbert space

A vector space over the field of complex or real numbers equipped with a scalar product , , satisfying the following conditions:

1) , , , , ();

2) for ;

3) if and only if .

On a pre-Hilbert space a norm is defined. The completion of with respect to this norm is a Hilbert space.

Comments

A function 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 is a normed linear space, then it has an inner product generating the norm if (and only if) the norm satisfies the parallelogram law

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