1.2.14 Dual Vector Space Definition

Dual Vector Space Definition is the characterization of the dual space of a vector space as the set of all linear functionals defined on that space — all linear maps from the vector space to its underlying scalar field — equipped with its own addition and scalar multiplication operations that make it a vector space in its own right. It supplies the second fundamental vector space associated with any given vector space, and it is an indispensable ingredient in the general definition of a tensor, since tensors of mixed type are built from tensor products involving both a vector space and its dual.

Constructing the Dual Space

Given a vector space over a scalar field, a linear functional on that space is a linear map sending vectors to scalars. The collection of all such linear functionals, taken together, forms the dual space of the original vector space. This dual space is itself a vector space: two linear functionals can be added by adding their outputs for every input vector, and a linear functional can be scaled by multiplying its output by a fixed scalar, and these operations satisfy the vector space axioms because the scalar field's own arithmetic satisfies them.

The expression above defines the dual space of a vector space as the set of all linear maps from that vector space into its underlying scalar field.

Elements of the Dual Space: Covectors

Elements of the dual space are called covectors, dual vectors, or one-forms, depending on the context in which they are used. A covector takes a single vector as input and produces a scalar, and it does so linearly, meaning it respects sums and scalar multiples of its input. Where a vector can be pictured, informally, as an object with a magnitude and direction, a covector is better understood functionally, as a measuring device that assigns a number to every vector in a manner compatible with the vector space's linear structure.

The Dual Basis

If the original vector space is finite-dimensional and a basis has been chosen for it, a corresponding dual basis can be constructed for the dual space, consisting of one covector for each basis vector, defined so that each dual basis covector returns one when applied to its corresponding basis vector and zero when applied to any other basis vector. This construction guarantees that the dual space has the same dimension as the original vector space, and it establishes the natural pairing between a vector and a covector, obtained by applying the covector to the vector, that recurs throughout tensor algebra as the operation of contraction between an upper and a lower index.

The Double Dual

Applying the dual space construction a second time, to the dual space itself, produces the double dual of the original vector space. For finite-dimensional vector spaces, there is a natural, basis-independent identification between a vector space and its double dual, obtained by associating to each vector the linear functional on the dual space given by evaluating covectors at that vector. This natural identification is what justifies treating a vector, in the context of tensor algebra, as itself a kind of linear functional acting on covectors, placing vectors and covectors on a symmetric footing within the broader tensor formalism.

Role in Tensor Algebra

The dual space is essential to the general definition of a tensor, since a tensor of type (p, q) is defined as an element of a tensor product involving p copies of the original vector space and q copies of its dual space. Covariant tensor indices correspond to factors drawn from the dual space, while contravariant indices correspond to factors drawn from the original space, and the distinction between how these two kinds of indices transform under a change of basis is a direct consequence of how the dual basis is constructed from the original basis. Without the dual space, only tensors built from a single vector space could be defined, excluding covectors, linear functionals, and every mixed tensor from the theory entirely.