V and a canonical cobasis {εi} for V∗, a -dimensional array of components can be obtained. A different choice of basis will yield different components. But, because T is linear in all of its arguments, the components satisfy the tensor transformation law used in the multilinear array definition. The multidimensional array of components of T thus form a tensor according to that definition. Moreover, such an array can be realized as the components of some multilinear map T. This motivates viewing multilinear maps as the intrinsic objects underlying tensors. In viewing a tensor as a multilinear map, it is conventional to identify the double dual V∗∗ of the vector space V, i.e., the space of linear functionals on the dual vector space V∗, with the vector space V. There is always a natural linear map from V to its double dual, given by evaluating a linear form in V∗ against a vector in V. This linear mapping is an isomorphism in finite dimensions, and it is often then expedient to identify V with its double dual. Using tensor products For some mathematical applications, a more abstract approach is sometimes useful. This can be achieved by defining tensors in terms of elements of tensor products of vector spaces, which in turn are defined through a universal property as explained here and here. A type tensor is defined in this context as an element of the tensor product of vector spaces, A basis of and basis of naturally induce a basis of the tensor product . The components of a tensor are the coefficients of the tensor with respect to the basis obtained from a basis for and its dual basis , i.e. Using the properties of the tensor product, it can be shown that these components satisfy the transformation law for a type tensor. Moreover, the universal property of the tensor product gives a one-to-one correspondence between tensors defined in this way and tensors defined as multilinear maps. This 1 to 1 correspondence can be archived the following way, because in the finite dimensional case there exists a canonical isomorphism between a vectorspace and its double dual: The last line is using the universal property of the tensor product, that there is a 1 to 1 correspondence between maps from and . Tensor products can be defined in great generality – for example, involving arbitrary modules over a ring. In principle,