Dual Vector Space

The dual vector space of a vector space over a field is the vector space of linear transformations , with pointwise addition and scalar multiplication (Axler 2026, p. 105).

For a real vector space, these linear functionals take values in . For a complex vector space, they take values in the complex numbers.

This is the algebraic dual, which imposes no continuity requirement. For a real or complex normed space, the continuous linear functionals instead form its dual normed space. In finite dimensions, every linear functional is continuous for any norm on . The two duals can differ in infinite dimensions.

If is finite-dimensional, its 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 (Axler 2026, pp. 105-107).

For finite-dimensional , 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 vector 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. With the convention that it is linear in its first argument, gives an antilinear bijection from onto , so (Axler 2026, p. 216). These identifications depend on the chosen inner product.

In contrast, evaluation defines a canonical linear transformation by

Its definition requires neither a vector basis nor an inner product. For finite-dimensional , this map is an isomorphism (Axler 2026, p. 118).

Dual vector spaces can describe many objects in linear algebra. When and are finite-dimensional vector spaces, an element of the vector space tensor product , say , corresponds to the linear transformation . That is, . For example, the identity map corresponds to . A bilinear form on , such as a real inner product, is an element of . A complex Hermitian inner product is a sesquilinear form rather than complex bilinear.