on V. As a special case, note that if F is a linear form (or (0,1)-tensor) on W, so that F is an element of W^*, the dual space of W, then ?^*F is an element of V^*, and so pullback by ? defines a linear map between dual spaces which acts in the opposite direction to the linear map ? itself:

From a tensorial point of view, it is natural to try to extend the notion of pullback to tensors of arbitrary rank, i.e., to multilinear maps on W taking values in a tensor product