In modern geometry, Euclidean spaces are often defined from linear algebra. More precisely, a Euclidean space E is defined as a set to which is associated an inner product space of finite dimension over the reals E → → , and a group action of the additive group of E → → , which is free and transitive (See Affine space for details of this construction). The elements of E → → are called translations. It has been proven that the two definitions of Euclidean spaces are equivalent, and that the equivalence classes under equipollence may be identified with translations.