Tensor product of Hilbert spaces

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. Roughly speaking, the tensor product is the metric space completion of the ordinary tensor product. This is an example of a topological tensor product. The tensor product allows Hilbert spaces to be collected into a symmetric monoidal category.

Since Hilbert spaces have inner products, one would like to introduce an inner product, and therefore a topology, on the tensor product that arise naturally from those of the factors. Let H₁ and H₂ be two Hilbert spaces with inner products ${\displaystyle \langle \cdot ,\cdot \rangle _{1}}$ and ${\displaystyle \langle \cdot ,\cdot \rangle _{2}}$, respectively. Construct the tensor product of H₁ and H₂ as vector spaces as explained in the article on tensor products. We can turn this vector space tensor product into an inner product space by defining

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