Editing Dual basis (section)

==Existence and uniqueness==
The dual set always exists and gives an injection from ''V'' into ''V''<sup>∗</sup>, namely the mapping that sends ''v<sub>i</sub>'' to ''v<sup>i</sup>''. This says, in particular, that the dual space has dimension greater or equal to that of ''V''.

However, the dual set of an infinite-dimensional ''V'' does not span its dual space ''V''<sup>∗</sup>. For example, consider the map ''w'' in ''V''<sup>∗</sup> from ''V'' into the underlying scalars ''F'' given by {{nowrap|1=''w''(''v<sub>i</sub>'') = 1}} for all ''i''. This map is clearly nonzero on all ''v<sub>i</sub>''. If ''w'' were a finite [[linear combination]] of the dual basis vectors ''v<sup>i</sup>'', say <math display="inline">w=\sum_{i\in K}\alpha_iv^i</math> for a finite subset ''K'' of ''I'', then for any ''j'' not in ''K'', <math display="inline">w(v_j)=\left(\sum_{i\in K}\alpha_iv^i\right)\left(v_j\right)=0</math>, contradicting the definition of ''w''. So, this ''w'' does not lie in the span of the dual set.

The dual of an infinite-dimensional space has greater dimension (this being a greater infinite cardinality) than the original space has, 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 space]]s, 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 <math>B=\{e_1,\dots,e_n\}</math> and <math>B^*=\{e^1,\dots,e^n\}</math>. If one denotes the evaluation of a covector on a vector as a pairing, the biorthogonality condition becomes:
:<math>\left\langle e^i, e_j \right\rangle = \delta^i_j.</math>

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''<sup>∗</sup>, and this is also an isomorphism. For [[topological field]]s such as the real numbers, the space of duals is a [[topological space]], and this gives a [[homeomorphism]] between the [[Stiefel manifold]]s of bases of these spaces.