One can think of the tensor product of two vector spaces, formula_8 and formula_9, as representing the set of all tensors that take a vector from formula_8 and a vector from formula_9 and output a scalar within their common base field (and thus can only be defined if they have such a common base field). The two spaces may be the same—above, they are vectors in the "tangent space" at a point: roughly the flat space a tiny piece of the manifold "looks like" near a particular point, and thus the metric tensor lives in the tensor product of that space with itself. But the two spaces may also be different.