Definition
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 and, 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
and extending by linearity. That this inner product is the natural one is justified by the identification of scalar-valued bilinear maps on H1 × H2 and linear functionals on their vector space tensor product. Finally, take the completion under this inner product. The resulting Hilbert space is the tensor product of H1 and H2.
Read more about this topic: Tensor Product Of Hilbert Spaces
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