3.2.1 Tensor Covector Domain Relation

Tensor Covector Domain Relation is the account of how a covector's domain, the vector space V it consumes vectors from, governs the structure of the covector itself, covering how the dimension and basis of V fix the dimension and dual basis of V*, how a linear map on the domain side induces a corresponding pullback on covectors, and how the domain relation is what makes a covector meaningfully attached to one particular vector space rather than being a free-floating linear functional with no fixed source.

The Domain Fixes the Covector Space

V* Is Defined Relative to V

A covector is never defined in isolation; ω ∈ V* is meaningful only relative to a specific domain V, since V* is by definition the space of linear functionals V → F. Two different domains V and U, even of equal dimension, produce different dual spaces V* and U*, and a covector belonging to one is not automatically a covector belonging to the other without an explicit identification of V with U.

Dimension Inherited From the Domain

Since dim(V*) = dim(V), the domain determines the dimension of the covector space exactly; an n-dimensional domain produces an n-dimensional space of covectors, with the dual basis {e^1, ..., e^n} inheriting one basis covector for each basis vector of the domain.

Change of Domain Basis and Its Effect on Covectors

Dual Basis Transformation

If the domain basis changes from {e_i} to {e'_i} = Σ_j a^j_i e_j, the dual basis of V* transforms with the inverse matrix, e'^i = Σ_j b^i_j e^j, so a change originating entirely in the domain propagates directly into a corresponding, fully determined change in how covectors are described in coordinates; no independent choice is available on the covector side once the domain basis is fixed.

Covector Components Transform Covariantly

The components ω_i of a covector transform with the direct matrix A, as already established in dual spaces and covectors, precisely because this is the transformation rule required for the pairing ω(v) = Σ_i ω_i v^i to remain independent of the domain basis; the domain relation is what forces covector components to transform covariantly rather than contravariantly, since it is the domain's contravariant vector components that must be compensated.

Domain Maps Induce Covector Maps in Reverse

The Pullback Construction

Given a linear map φ : V → U between domains, the pullback φ* : U* → V*, defined by φ*(η) = η ∘ φ, sends a covector on the target domain U back to a covector on the source domain V. This is the direct, single-covector instance of the direction-reversing carrier functoriality established for dual tensor spaces generally: because covectors are attached to their domain, moving a covector to a different domain requires composing with a map that produces elements of the covector's original domain, which is exactly what φ* does when φ maps into U.

Restriction to a Subspace as a Domain Relation

When V is a subspace of a larger space U, the inclusion map ι : V → U induces a restriction map ι* : U* → V* sending a covector defined on all of U to its restriction on V, ι*(η) = η|_V. This is a concrete, commonly used instance of the domain relation: restricting the domain of interest correspondingly restricts, via pullback, the covectors under consideration.

Domain Relation and the Natural Pairing

The Pairing Is Domain-Specific

The natural pairing V* × V → F sending (ω, v) to ω(v) is defined using precisely the domain V that ω is dual to; attempting to pair ω ∈ V* with a vector from a different space U ≠ V is not meaningful unless an explicit map between U and V is supplied first. The domain relation is therefore not an incidental labeling convention but the essential fact that determines which vectors a given covector is capable of pairing with at all.

Consistency With Multilinear Functional Interpretation

For higher tensors built with lower indices drawn from V*, each lower index slot inherits the same domain relation to V: a (p, q)-tensor accepts exactly q vector arguments from V, not from any other space, because each of its q lower-index factors is a covector on V specifically, carrying forward the domain relation established for a single covector to the fully general multilinear-functional picture of tensors.