and thus these cannot have a basis with the same indexing set. However, a dual set of vectors exists, which defines a subspace of the dual isomorphic to the original space. Further, for topological vector spaces, a continuous dual space can be defined, in which case a dual basis may exist. Finite-dimensional vector spaces In the case of finite-dimensional vector spaces, the dual set is always a dual basis and it is unique. These bases are denoted by and . If one denotes the evaluation of a covector on a vector as a pairing, the biorthogonality condition becomes: The association of a dual basis with a basis gives a map from the space of bases of V to the space of bases of V∗, and this is also an isomorphism. For topological fields such as the real numbers, the space of duals is a topological space, and this gives a homeomorphism between the Stiefel manifolds of bases of these spaces. A categorical and algebraic construction of the dual space Another way to introduce the dual space of a vector space (module) is by introducing it in a categorical sense. To do this, let be a module defined over the ring (that is, is an object in the category ). Then we define the dual space of , denoted , to be , the module formed of all -linear module homomorphisms from into . Note then that we may define a dual to the dual, referred to as the double dual of , written as , and defined as . To formally construct a basis for the dual space, we shall now restrict our view to the case where is a finite-dimensional free (left) -module, where is a ring with unity. Then, we assume that the set is a basis for . From here, we define the Kronecker Delta function over the basis by if and if . Then the set describes a linearly independent set with each . Since is finite-dimensional, the basis is of finite cardinality. Then, the set is a basis to and is a free (right) -module. Examples For example, the standard basis vectors of R2 (the Cartesian plane) are and the standard basis vectors of its dual space R2* are In 3-dimensional Euclidean space, for a given basis {e1, e2, e3}, the biorthogonal (dual) basis {e1, e2, e3} can be found by formulas below: where denotes