3.3.4 Tensor Linear Functional Scalar Preservation

Tensor Linear Functional Scalar Preservation is the fact that when a linear functional is scaled by the pointwise scalar-multiplication operation defining the vector-space structure of V*, the resulting function remains a linear functional, satisfying both membership conditions required for inclusion in V*, so that V* is closed under scalar multiplication in exactly the sense needed for it to qualify as a vector space, paralleling the closure under addition established by additivity preservation.

Statement of the Preservation Property

The Scalar Multiple in Question

For ω ∈ V* and a scalar d ∈ F, the scalar multiple dω is defined pointwise by (dω)(v) = d(ω(v)) for every v ∈ V. Scalar preservation is the claim that dω, so defined, itself satisfies the membership test for V*: it must be additive, (dω)(v_a + v_b) = (dω)(v_a) + (dω)(v_b), and homogeneous, (dω)(cv) = c(dω)(v).

Verification of the Additive Condition

Direct Computation

For v_a, v_b ∈ V:

using additivity of ω itself, and then distributing d over the sum, d(ω(v_a) + ω(v_b)) = d ω(v_a) + d ω(v_b) = (dω)(v_a) + (dω)(v_b), using distributivity of multiplication over addition in the field F. This confirms the additive half of the membership test for dω.

Verification of the Homogeneous Condition

Direct Computation

For v ∈ V and a scalar c:

using homogeneity of ω, then associativity of multiplication in F to regroup d and c, then commutativity of multiplication in F to swap their order, arriving at exactly the homogeneity condition required of dω. Both the associativity and commutativity of the field's multiplication are essential to this argument, not merely convenient.

Closure of V* Under Scalar Multiplication

V* as Closed Under the Pointwise Scalar Action

Because both conditions hold for arbitrary ω ∈ V* and d ∈ F, the set V* is closed under pointwise scalar multiplication: a scalar multiple of a linear functional is always another linear functional. Together with additivity preservation, this closure supplies the two structural facts needed for V*, under its pointwise operations, to satisfy the closure axioms of a vector space, prior to checking the remaining associativity, identity, and distributivity axioms, which follow immediately from the corresponding axioms already holding pointwise in F.

Behavior at the Scalars Zero and One

Scalar preservation holds in particular for the boundary cases d = 0, giving the zero functional, and d = 1, giving ω back unchanged, both trivially satisfying the membership test; these boundary cases confirm that scalar preservation is consistent with V* containing an additive identity and with scalar multiplication by 1 acting as the identity operation, two further requirements of the vector-space axioms.

Interaction With Kernel and Rank Under Scaling

Kernel Invariance for Nonzero Scalars

As already noted in tensor linear functional kernel structure, ker(dω) = ker(ω) whenever d ≠ 0, since scalar preservation guarantees dω remains a well-defined functional whose zero set is unaffected by a nonzero rescaling; only d = 0 collapses dω to the zero functional, whose kernel is all of V, rather than the codimension-one kernel of a nonzero ω.