Linear maps and tensor rank

Let V 1 , …, V m be inner product spaces and A a linear operator on V 1 ⊗ ··· ⊗ V m . Suppose that an equation involving A holds for all tensors of a given rank. Does it follow that the equation holds for all tensors ? We answer this question for some equations involving the inner production V 1 ⊗ ··· ⊗ V m . For example, it is shown that if the field is the complex numbers and ( At , t ) = 0, for all decomposable tensors t , then ( At , t ) = 0, for all tensors t . Thus, A = 0.