Duality

Dual Space & Covector

The dual space of a vector space , denoted as , is the set of all linear maps from to the underlying field . i.e. . And elements of are called covectors, dual vectors or linear functionals.

Proposition

The dual space is a vector space.

Proof It is clear enough to show that both associativity, commutativity hold. We identify the constant function as the additive identity.

Proposition

Suppose is a basis for . Then defined by forms a basis for .

Proof

Corollary

For any vector space , , and further .

Proof This a consequence of the above proposition, because is a linear isomorphism.

Thrm In a inner product space , for any , there is a unique such that .
Proof We identify any by identifying its values on the basis vectors. Suppose form a basis for . Then we have

Proposition

There is a natural isomorphism between and . That is, the double dual functor is naturally isomorphic to the identity functor on the category of vector spaces.

Proof Define the natural transformation by We shall check the following diagram commutes for any :

For any and , there holds Thus the diagram commutes, and is a natural transformation. Moreover, is an isomorphism for any , thus is a natural isomorphism.

Geometric Interpretation

Geometrically, this definition uses no bases or coordinates—it is intrinsically defined by the structure of vector spaces. That’s why “natural” is often equated with basis-independence or coordinate-free definition in geometry. Concretely, if one identify as the coordinate, then the above commutative diagram basically means coordinate free.