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I don't quite understand how tensor products of Hilbert spaces are formed.

What I get so far is that from two Hilbert spaces [itex]\mathscr{H}_1[/itex] and [itex]\mathscr{H}_2[/itex] a tensor product [itex]H_1 \otimes H_2 [/itex] is formed by considering the Hilbert spaces as just vector spaces [itex]H_1[/itex] and [itex]H_2[/itex].

Next, there is an inner product on this space, which is defined by

[itex]\langle \phi_1 \otimes \phi_2 \vert \psi_1 \otimes \psi_2 \rangle \equiv \langle \phi_1 \vert \psi_1 \rangle_1 \langle \phi_2 \vert \psi_2 \rangle_2 [/itex] on the simple or pure tensors on this tensor product space. This inner product is extended linearly to an inner product on all elemnets of the tensor product space.

This is where my understanding stops. The 'completion' of this tensor product space is now taken, and the result is a Hilbert space, which is then defined as the tensor product of the Hilbert spaces.

This seems weird to me, because it seems artificial - is the tensor product of Hilbert spaces defined such that its a Hilbert space again?

I don't really understand how taking the completion of a space works. Can anyone provide some insight as to how this works?

Thanks for any help

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# Tensor product of Hilbert spaces

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