3.3.1 Tensor Linear Functional Domain Space

Tensor Linear Functional Domain Space is the vector space V that a linear functional ω : V → F takes its input from, examined specifically for how the domain governs the general functional-theoretic facts established in tensor linear functional structure: how the dimension of the domain fixes the codimension of a functional's kernel, how restricting a functional to a subspace of the domain relates to the annihilator construction, and how the domain determines which subspaces are even candidates for a given functional's kernel or annihilator.

Domain Dimension and Kernel Codimension

The Governing Formula

For a nonzero functional ω on domain V, the kernel identity dim(ker(ω)) = dim(V) - 1 is a direct function of the domain's dimension: doubling the dimension of the domain, while keeping ω nonzero, doubles neither the kernel's dimension nor the codimension, since codimension is fixed at exactly 1 regardless of dim(V), so long as dim(V) ≥ 1.

Degenerate Case of a Zero-Dimensional Domain

If dim(V) = 0, then V = {0} and the only functional on V is the zero functional, since there is no nonzero vector to assign a nonzero value to consistently with linearity; the kernel-codimension formula is vacuous in this degenerate case, and functional structure becomes meaningful only once the domain has dimension at least 1.

Restriction of a Functional to a Subspace of the Domain

The Restriction Map

Given a subspace U ⊆ V and a functional ω ∈ V*, the restriction ω|_U : U → F is again a linear functional, now with the smaller domain U. This restriction is exactly the pullback ι*(ω) along the inclusion map ι : U → V, and it is how a single functional on a large domain produces a corresponding functional on any smaller domain contained within it.

Relation to the Annihilator

The functional ω restricts to the zero functional on U, ω|_U = 0, precisely when ω ∈ U°, the annihilator of U; the domain-restriction operation and the annihilator construction are two views of the same underlying fact, that ω vanishes on every vector of the smaller domain U. This connects the domain of a functional directly to the subspace-annihilator relationship: annihilation is nothing more than the statement that a functional's restriction to a given domain is identically zero.

Domain-Dependence of Kernel Candidacy

Not Every Subspace Is a Kernel

A subspace W ⊆ V arises as ker(ω) for some nonzero ω ∈ V* if and only if dim(W) = dim(V) - 1; subspaces of any other dimension cannot be the kernel of a nonzero functional on that particular domain V, since the kernel-codimension formula fixes codimension at exactly 1. Whether a given subspace can serve as a kernel is therefore a question that depends jointly on the subspace and on the specific domain V it sits inside.

Same Subspace, Different Domains

A fixed subspace W of dimension k may be a valid kernel candidate relative to one ambient domain V, with dim(V) = k + 1, but fail to be a valid kernel candidate relative to a larger domain V' ⊃ V with dim(V') > k + 1; kernel candidacy is a relation between a subspace and its specific ambient domain, not an intrinsic property of the subspace alone.

Domain and the Extension of a Functional

Extending From a Subspace Back to the Full Domain

Given a functional ψ defined only on a subspace U ⊆ V, it is always possible to extend ψ to a functional ω on the full domain V satisfying ω|_U = ψ, by choosing a complementary subspace W with V = U ⊕ W and defining ω to agree with ψ on U and to take arbitrary, for instance zero, values on W. This extension is generally not unique, since different choices of complement W or different values assigned on W produce different extensions ω, all agreeing with ψ when restricted back down to the original smaller domain U.

Contrast With Restriction

Restriction from V down to U is a single, well-defined operation, producing exactly one functional ω|_U from a given ω, whereas extension from U up to V is a one-to-many process; this asymmetry mirrors the general asymmetry of the annihilator and pullback constructions, in which passing to a smaller domain is unambiguous while passing to a larger domain requires an additional, non-canonical choice.