# Identification of products in the tensor product of hilbert spaces

I’m relatively new to tensor products and particularly interested in tensor products of Hilbert spaces. I did read a short note on the explicit construction of theses spaces, which is also covered on Wikipedia (link below). However, I have some general question and hope someone can bring some light into my confusion.

Following the wikipedia article one can construct the tensor product of Hilbert spaces $$H_1$$ and $$H_2$$ as the space which is isometrically and linearly isomorphic to $$HS(H_{1}^{*},H_{2})$$, the space of Hilber-Schmidt operators from $$H_1^*$$ to $$H_2$$.

The idea is to identify to every tensor $$x_1otimes x_2$$ with $$x_iin H_i$$ and $$x^*in H_1^*$$ the map

$$x^*mapsto x^*(x_1)x_2$$

On the other hand one often defines the tensor product via a new inner product. For this the tensor product (in algebraic way) is first constructed. Then an inner product is defined. I quote Wikipedia:

Construct the tensor product of $$H_1$$ and $$H_2$$ 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
$${displaystyle langle phi _{1}otimes phi _{2},psi _{1}otimes psi _{2}rangle =langle phi _{1},psi _{1}rangle _{1},langle phi _{2},psi _{2}rangle _{2}quad {mbox{for all }}phi _{1},psi _{1}in H_{1}{mbox{ and }}phi _{2},psi _{2}in H_{2}}$$

and extending by linearity. That this inner product is the natural one is justified by the identification of scalar-valued bilinear maps on $$H_1 times H_2$$ 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 $$H_1$$ and $$H_2$$.

question 1 In the last definition, if I read it correctly, we build first the tensor product in a algebraic way. Then an inner product is on that tensor product defined. Finally the completion is taken and one sees that this corresponds with the original (algebraic) tensor product? Or how does the algebraic and the one stemming from the completion related?

question 2 I often see that for Hilbert spaces, say of functions, tensor products are defined via multiplication. That is for functions $$f_iin H_i$$ we have $$f_1otimes f_2:=f_1(x)f_2(x)$$. How do we get this concrete example of “multiplication”. I’m particular interested in Hilbert spaces of scalar valued functions, if that matters.

question 3. What is the relation of the explicit construction via Hilber-Schmidt operators to the multiplicative case mentioned in question 2.

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