Abelian group
In other words, each group element formula_21 is identified to the evaluation character on the dual. This is strongly analogous to the canonical isomorphism between a finite-dimensional vector space and its double dual, formula_22, and it is worth mentioning that any vector space formula_23 is an Abelian group. If formula_2 is a finite abelian group, then formula_25 but this isomorphism is not canonical. Making this statement precise (in general) requires thinking about dualizing not only on groups, but also on maps between the groups, in order to treat dualization as a functor and prove the identity functor and the dualization functor are not naturally equivalent. Also the duality theorem implies that for any group (not necessarily finite) the dualization functor is an exact functor.