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|
Pre-Hilbert space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pre-Hilbert_space&oldid=15523