The dual vector space to a real vector space is the vector space of linear functions , denoted . In the dual of a complex vector space, the linear functions take complex values.
In either case, the dual vector space has the same dimension as . Given a vector basis , ..., for there exists a dual basis for , written , ..., , where and is the Kronecker delta.
Another way to realize an isomorphism with is through an inner product. A real vector space can have a symmetric inner product in which case a vector corresponds to a dual element by . Then a basis corresponds to its dual basis only if it is an orthonormal basis, in which case . A complex vector space can have a Hermitian inner product, in which case is a conjugate-linear isomorphism of with , i.e., .
Dual vector spaces can describe many objects in linear algebra. When and are finite dimensional vector spaces, an element of the tensor product , say , corresponds to the linear transformation . That is, . For example, the identity transformation is . A bilinear form on , such as an inner product, is an element of .