Pre-Hilbert space

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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.


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.


[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. V.I. Lomonosov (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098