3.2.3 Tensor Covector Linearity Structure

Tensor Covector Linearity Structure is the foundational requirement that a covector, as a function ω : V → F, satisfy the two defining linearity conditions, additivity and homogeneity, in its single vector argument, together with the consequences these conditions have for how a covector interacts with linear combinations of vectors, how it is determined by its values on a basis, and how linearity in this single-argument sense is the base case from which the multilinear behavior of every higher tensor slot is built.

The Two Linearity Conditions

Additivity

A covector ω satisfies additivity if, for all v_1, v_2 ∈ V:

meaning the value of ω on a sum of two vectors is the sum of its values on each vector individually; a covector cannot introduce any interaction term between v_1 and v_2 beyond the ordinary sum of independent contributions.

Homogeneity

A covector ω satisfies homogeneity if, for all v ∈ V and scalars c:

meaning scaling the input by c scales the output by the same c; a covector's response to a rescaled vector is exactly proportional, with no higher-order dependence on c.

Determination by Values on a Basis

Uniqueness From Basis Values

Because of additivity and homogeneity, the value of ω on any v = Σ_i v^i e_i, written relative to a basis {e_i} of V, is completely determined by the values ω(e_1), ..., ω(e_n):

so a covector is uniquely and completely specified by the finite list of scalars ω_i = ω(e_i); no additional information beyond these n numbers is needed to reconstruct ω's action on every vector in V.

Freedom to Prescribe Basis Values Arbitrarily

Conversely, any assignment of n arbitrary scalars to ω(e_1), ..., ω(e_n) extends, via the formula above, to a well-defined linear functional on all of V, with no consistency conditions to check beyond the formula itself; this freedom is what makes the dual basis construction, e^i(e_j) = δ^i_j, a valid definition rather than an overdetermined or inconsistent one, and it is what gives dim(V*) = dim(V) its concrete, basis-level justification.

Linearity Structure as the Base Case for Multilinearity

Single-Slot Linearity Extends to Multi-Slot Multilinearity

The linearity conditions for a single covector are the q = 1 case of the general multilinearity required of a (0, q)-tensor: a (0, q)-tensor is required to be linear in each of its q vector arguments separately, holding the other q - 1 arguments fixed, and this reduces, when q = 1, exactly to the two conditions above. The covector slot role described elsewhere, reading a single designated argument, is compatible with linearity precisely because that argument enters through this same single-slot linearity condition.

What Linearity Rules Out

Linearity structure excludes any functional that responds to a vector argument in a way involving powers, products of components, or other nonlinear dependence, such as v ↦ (v^1)^2 or v ↦ v^1 v^2; such functionals are not covectors, and do not belong to V*, even though they may be well-defined functions V → F. The dual space V* is, by the linearity structure required of its elements, strictly smaller than the space of all functions from V to F, containing only those functions compatible with the vector-space operations of V.

Consistency With Scalar Action and Additivity of V* Itself

Linearity of a Covector Versus Vector-Space Structure of V*

The linearity conditions on an individual covector ω, concerning how ω acts on sums and scalar multiples of vectors in V, are logically distinct from the vector-space operations on V* itself, concerning how two covectors ω_1 and ω_2 are added or scaled as elements of V*; the first is a property required of each element of V*, and the second is the structure making the collection of all such elements into a vector space. Both layers of linearity are necessary for the dual space construction to function: without the first, V* would not consist of linear functionals at all, and without the second, V* would not itself be a vector space.