Transformations on tensor spaces II

We prove that a linear map of one tensor product space to another sending decomposable tensors to decomposable tensors is essentially a tensor product of linear maps of products of component factors of the domain into a selection of the factors of the range. The product of those factors of the domain not involved in the above is collapsed via a linear functional, and those factors in the range left out of the above provide a common factor in the range. In the statement of the main theorem the flanking maps are induced by permutations of the factors of the domain and the range and they present the products in manageable form. It is assumed that the underlying field has at least five members, but the necessity of this assumption is not settled. All vector spaces are finite dimensional and the tensor products have finitely many components.