In mathematics, and in particular functional analysis, the tensor product of Hilbert spaces is a way to extend the tensor product construction so that the result of taking a tensor product of two Hilbert spaces is another Hilbert space.

In mathematics, specifically in functional analysis and Hilbert space theory, the fundamental theorem of Hilbert spaces gives a necessarily and sufficient condition for a Hausdorff pre-Hilbert space to be a Hilbert space in terms of the canonical isometry of a pre-Hilbert space into its anti-dual.