H1 and H2 are two Hilbert spaces represented by a function space, say f1(x1) and f2(x2) are its vectors. If H3 is tensor product of H1 and H2 I assume one can say that f(x1,x2) now represents vectors in H3.
The scalar product for f(x1,x2) would be a tensor scalar product.
Now say that f(x1,x2) can be factorized into f1(x1)f2(x2).
Does that mean that this vector is in the Cartesian product subspace of the tensor product space and that scalar product of f1(x1)f2(x2) would be identical to a scalar product as calculated for Cartesian product spaces?
Edit: I am not even sure that the concept of Cartesian product subspace of the tensor product space makes sense.