3.2.2 Tensor Covector Scalar Codomain

Tensor Covector Scalar Codomain is the fact that a covector is, by definition, a linear map whose codomain is the underlying scalar field F itself rather than an arbitrary vector space W, and the collection of consequences this special choice of codomain has for the structure of V*, including the identification of V* with Hom(V, F), the collapse of the general Hom(V, W) ≅ V* ⊗ W identification to the simpler V* ≅ V* ⊗ F, and the resulting absence of any additional upper-index tensor factor in the description of a covector, since the scalar field, treated as a one-dimensional vector space over itself, contributes nothing beyond a single trivial basis vector.

The Scalar Field as a Special Codomain

Covectors as Hom(V, F)

A covector ω ∈ V* is, by the original definition of the dual space, a linear map ω : V → F, so V* = Hom(V, F) by definition rather than by any further argument. The scalar field F, viewed as a vector space over itself, is one-dimensional, with any nonzero element, most naturally 1, serving as a basis.

Consequence for the General Identification

Applying the general identification Hom(V, W) ≅ V* ⊗ W, described in tensor linear map structure, to the special case W = F gives:

since tensoring any vector space with the one-dimensional space F produces a space canonically isomorphic to the original space, V* ⊗ F ≅ V* via ω ⊗ c ↦ cω. The general two-factor identification therefore degenerates, in the scalar-codomain case, to the identity map on V* itself, confirming consistency rather than producing new information.

Why the Scalar Codomain Contributes No Extra Index

Comparison With a General Codomain W

For a general linear map φ : V → W, the matrix representation a^j_i has a lower index i, from the domain's dual, and an upper index j, from the codomain W. When W = F is one-dimensional, the codomain basis has only a single element, so the upper index j ranges over only one value and can be omitted entirely; the "matrix" of a covector is a single row, and is written simply as ω_i, a purely lower-index object, rather than as a^1_i with a redundant upper index fixed at 1.

Rank as a (0, 1)-Tensor

This is why a covector is classified as a (0, 1)-tensor, with zero upper indices, rather than a (1, 1)-tensor: the codomain of a covector is the scalar field, contributing a dim = 1 factor that is conventionally suppressed rather than tracked as a genuine tensor index. Every general linear map Hom(V, W) is a (1, 1)-tensor with respect to V and W jointly, but the special case W = F reduces this to a (0, 1)-tensor with respect to V alone.

Coordinate Consequences

Components as Ordinary Numbers

Because the scalar codomain contributes no basis choice beyond the trivial one, the components ω_i = ω(e_i) of a covector are simply elements of F, unadorned by any codomain-basis label, in contrast with the components a^j_i of a general linear map, which are indexed by both a domain-basis label i and a codomain-basis label j. This is a direct coordinate manifestation of the scalar codomain's dimensional triviality.

Full Contraction Yields the Scalar Codomain Directly

When a covector ω is fully contracted against a vector v, ω(v) ∈ F, the result lands directly in the scalar codomain itself, with no further tensor structure remaining; this is the simplest possible instance of a full tensor contraction, reducing a (0, 1)-tensor and a (1, 0)-tensor to a (0, 0)-tensor, which is by definition just a scalar, an element of F.

Relation to the General Multilinear Functional Picture

Consistency With T^p_q(V) as Functionals Into F

The identification of general (p, q)-tensors as multilinear functionals, described in tensor dual space structure, already targets the scalar field F as the common codomain for every tensor regarded as a functional; a covector's scalar codomain is simply the smallest, single-argument instance of this same general pattern, consistent with the convention that every tensor, of whatever rank, is ultimately valued in the scalar field once all of its arguments have been supplied.