3 Dual Spaces and Covectors

Dual Spaces and Covectors is the study of the vector space V* of linear functionals on a vector space V, together with its elements, called covectors or dual vectors, and the structural facts governing how V* relates to V: the existence of a dual basis, the matching of dimension in the finite-dimensional case, the natural pairing between V and V*, and the distinct, opposite way covector components transform under a change of basis compared to ordinary vector components. Dual spaces are the foundation on which the entire index-variance structure of tensor algebra is built, since every lower, covariant tensor index traces back to a factor of V*.

Definition of the Dual Space

Linear Functionals

A linear functional on V is a linear map ω : V → F from V to its underlying scalar field F, satisfying ω(v_1 + v_2) = ω(v_1) + ω(v_2) and ω(cv) = c ω(v) for all v_1, v_2, v ∈ V and scalars c. The set of all linear functionals on V, equipped with the pointwise operations (ω_1 + ω_2)(v) = ω_1(v) + ω_2(v) and (cω)(v) = c(ω(v)), forms a vector space, denoted V* and called the dual space of V.

Covectors

An element ω ∈ V* is called a covector, or dual vector, or one-form in some contexts, and its defining role is to consume a vector from V and return a scalar; a covector is a measuring device, mapping vectors to numbers, in contrast with an ordinary vector, which is the thing being measured.

The Dual Basis

Construction From a Basis of V

Given a basis {e_1, ..., e_n} of a finite-dimensional V, the dual basis {e^1, ..., e^n} of V* is the unique set of covectors satisfying:

where δ^i_j is the Kronecker delta, equal to 1 when i = j and 0 otherwise. Each e^i is defined by specifying its value on every basis vector of V and extending linearly, and this defines e^i uniquely because a linear map out of V is completely determined by its values on a basis.

Coordinates as Applications of Dual Basis Vectors

For any v = Σ v^i e_i ∈ V, applying the dual basis vector e^j gives e^j(v) = Σ_i v^i e^j(e_i) = Σ_i v^i δ^j_i = v^j, so e^j is exactly the linear functional that reads off the j-th coordinate of a vector relative to {e_i}. This is the concrete sense in which the dual basis is "dual" to the original basis: each dual basis vector extracts one coordinate.

Dimension and the Finite-Dimensional Isomorphism V ≅ V*

Equal Dimension

Since the dual basis {e^1, ..., e^n} has exactly as many elements as the basis {e_1, ..., e_n} of V, dim(V*) = dim(V) = n whenever V is finite-dimensional. Consequently, V and V* are isomorphic as abstract vector spaces, since dimension is a complete isomorphism invariant.

No Natural Isomorphism

Although V ≅ V*, there is no canonical, basis-independent isomorphism between them: any isomorphism V → V* constructed by matching e_i to e^i depends on the choice of basis {e_i}, and a different basis produces a different, generally unequal, isomorphism. This absence of a natural identification is a central reason for keeping V and V* conceptually and notationally distinct, using upper indices for V and lower indices for V*, rather than conflating the two spaces.

The Natural Pairing

Evaluation as a Bilinear Pairing

The map V* × V → F sending (ω, v) to ω(v) is bilinear, linear in ω for fixed v and linear in v for fixed ω, and is called the natural pairing between V* and V. Unlike an isomorphism between V and V*, this pairing requires no choice of basis; it is defined directly from the meaning of V* as the space of linear functionals on V.

The Double Dual

Applying the dual construction twice gives V** = (V*)*, the space of linear functionals on V*. There is a canonical, basis-independent isomorphism V → V** sending v to the evaluation functional ω ↦ ω(v), and in the finite-dimensional case this canonical map is an isomorphism, allowing V to be identified with V** without reference to any basis, in sharp contrast with the situation for V and V* themselves.

Covariant Transformation of Covector Components

Components of a Covector

Relative to the dual basis {e^i}, a covector ω has components ω_i = ω(e_i), so that ω = Σ ω_i e^i. These components are written with a lower index, marking them as covariant, in contrast with the upper-index, contravariant components v^i of an ordinary vector.

Transformation Rule

Under a change of basis e'_i = Σ_j a^j_i e_j of V, the dual basis transforms as e'^i = Σ_j b^i_j e^j, using the inverse matrix B = A^{-1}, and the components of a covector transform with the direct matrix A:

This is the opposite of the contravariant rule v'^i = Σ_j b^i_j v^j governing vector components, and the two rules are precisely complementary: they combine to guarantee that the scalar ω(v) = Σ_i ω_i v^i computed by the natural pairing does not depend on which basis is used to express ω and v.