# Is there a relation between Cartesian and tensor product of function spaces and function factorizability

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.

Edit: I am not even sure that the concept of Cartesian product subspace of the tensor product space makes sense.

Let's start with this bit. Given vector spaces (or Hilbert spaces) $$V$$ and $$W$$ there's a canonical map $$V\times W \rightarrow V \otimes W$$. But this isn't a linear map, and its image isn't a subspace. But its image is still a subset, which we call the set of simple tensors.

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?

This bit is all exactly right. A function $$f(x_1,x_2)$$ can be factored as $$f_1(x_1)f_2(x_2)$$ if and only if it lies in the image of the Cartesian product under the above map. That is to say the simple tensors are precisely the ones that factor.

• before I accept the answer, could you, please, point me to a book or online reference where this is a bit more elaborated, hopefully with examples? I googled a lot but could never find anything on this. – Tony Mar 28 '16 at 13:06
• Especially the scalar product. I have been unable to see how tensor scalar product, in the case of simple tensors, ends up looking like the Cartesian scalar product. – Tony Mar 28 '16 at 13:47