called the It is defined by grouping all occurring "factors" together: writing for an element of and for an element of the dual space, Picking a basis of and the corresponding dual basis of naturally induces a basis for (this basis is described in the article on Kronecker products). In terms of these bases, the components of a (tensor) product of two (or more) tensors can be computed. For example, if and are two covariant tensors of orders and respectively (i.e. and ), then the components of their tensor product are given by Thus, the components of the tensor product of two tensors are the ordinary product of the components of each tensor. Another example: let be a tensor of type with components and let be a tensor of type with components Then and Tensors equipped with their product operation form an algebra, called the tensor algebra. Evaluation map and tensor contraction For tensors of type there is a canonical evaluation mapdefined by its action on pure tensors: More generally, for tensors of type with , there is a map, called tensor contraction, (The copies of and on which this map is to be applied must be specified.) On the other hand, if is , there is a canonical map in the other direction (called the coevaluation map') where is any basis of and is its dual basis. This map does not depend on the choice of basis. The interplay of evaluation and coevaluation can be used to characterize finite-dimensional vector spaces without referring to bases. Adjoint representation The tensor product may be naturally viewed as a module for the Lie algebra by means of the diagonal action: for simplicity let us assume then, for each where is the transpose of , that is, in terms of the obvious pairing on There is a canonical isomorphism given by Under this isomorphism, every in may be first viewed as an endomorphism of and then viewed as an endomorphism of In fact it is the adjoint representation of Linear maps as tensors Given two finite dimensional vector spaces , over the same field , denote the dual space of as , and the -vector space of all linear maps from to as . There is an isomorphism, defined by an action of the pure tensor on an element of Its "inverse" can be defined using a basis and its dual basis as