3.3 Tensor Linear Functional Structure

Tensor Linear Functional Structure is the body of general facts about linear functionals as maps V → F considered in their own right, independent of any particular tensor construction, covering the kernel of a nonzero functional as a codimension-one subspace, the annihilator of a subspace as a subspace of the dual, and the relationship between a functional's zero set and the geometric hyperplanes it determines. This structure supplies the linear-algebraic backbone that later, more tensor-specific accounts of covectors and dual tensor spaces build upon.

Kernels of Linear Functionals

The Kernel Is a Hyperplane

For a nonzero functional ω ∈ V*, its kernel ker(ω) = {v ∈ V : ω(v) = 0} is a subspace of V of dimension dim(V) - 1, a hyperplane through the origin. This follows from the rank-nullity theorem applied to ω : V → F: since ω is nonzero, its image is all of the one-dimensional space F, so dim(ker(ω)) = dim(V) - dim(F) = dim(V) - 1.

Two Functionals With the Same Kernel

If ω_1 and ω_2 are nonzero functionals with ker(ω_1) = ker(ω_2), then ω_2 = c ω_1 for some nonzero scalar c; a hyperplane through the origin determines its defining functional uniquely up to a single nonzero scalar multiple, so the correspondence between nonzero functionals and hyperplanes is many-to-one in a precisely controlled way, with each hyperplane corresponding to exactly a one-dimensional family of functionals.

The Annihilator of a Subspace

Definition

For a subspace U ⊆ V, the annihilator U° ⊆ V* is the set of covectors vanishing on all of U:

U° is itself a subspace of V*, since a sum or scalar multiple of functionals vanishing on U still vanishes on U.

Dimension Formula

The annihilator satisfies dim(U°) = dim(V) - dim(U), complementary to the dimension of U within V. This can be seen by extending a basis {e_1, ..., e_k} of U to a basis {e_1, ..., e_n} of V: the covectors e^{k+1}, ..., e^n from the dual basis span U°, giving exactly n - k basis elements.

Double Annihilator

For finite-dimensional V, (U°)° = U under the canonical identification V** ≅ V, so annihilating twice returns the original subspace; the annihilator operation is an order-reversing bijection between subspaces of V and subspaces of V*, satisfying U_1 ⊆ U_2 ⇒ U_2° ⊆ U_1°.

Functionals as Coordinates on Affine Level Sets

Level Sets Are Parallel Hyperplanes

For a nonzero ω ∈ V* and a scalar c, the level set {v ∈ V : ω(v) = c} is always nonempty, since ω is surjective onto F whenever nonzero, and it forms an affine hyperplane parallel to ker(ω) = {v : ω(v) = 0}, obtained by translating ker(ω) by any single vector v_0 satisfying ω(v_0) = c.

Functionals as a Coordinate System Transverse to a Hyperplane

Fixing a basis {e_1, ..., e_k} of a hyperplane ker(ω) and extending by one vector e_{k+1} with ω(e_{k+1}) = 1, every vector v ∈ V decomposes uniquely as v = u + t e_{k+1} for u ∈ ker(ω) and t = ω(v) ∈ F; the functional ω supplies exactly the single coordinate needed to measure displacement transverse to its own kernel, while the kernel itself supplies the remaining coordinates.

Relation to Tensor-Specific Covector Structure

Functional Structure as the Substrate for Covector Structure

The kernel and annihilator facts established here concern V* purely as a space of linear functionals, using only the rank-nullity theorem and basis extension, with no reference to the tensor product or to higher tensor constructions; the tensor-specific properties of covectors, their role as lower-index tensor slots, their transformation rule under change of basis, their combination via the tensor product, are additional structure layered on top of this more basic linear-functional foundation, and depend on it rather than replacing it.