3.3.3 Tensor Linear Functional Additivity Preservation

Tensor Linear Functional Additivity Preservation is the fact that when two linear functionals are combined by the pointwise addition operation defining V* as a vector space, the resulting function is itself still a linear functional, satisfying both membership conditions of additivity and homogeneity required for inclusion in V*, so that the set of linear functionals is closed under addition rather than merely being a set that happens to contain some linear and some nonlinear functions after combination.

Statement of the Preservation Property

What Must Be Verified

Given ω_1, ω_2 ∈ V*, their pointwise sum is defined by (ω_1 + ω_2)(v) = ω_1(v) + ω_2(v) for every v ∈ V. Additivity preservation is the claim that this sum, ω_1 + ω_2, itself satisfies the membership test for V*: it must be additive in its own argument, (ω_1 + ω_2)(v_a + v_b) = (ω_1 + ω_2)(v_a) + (ω_1 + ω_2)(v_b), and homogeneous, (ω_1 + ω_2)(cv) = c(ω_1 + ω_2)(v). Without this verification, the pointwise sum of two covectors would be merely some function V → F, not necessarily a new element of V*.

Verification of the Additive Condition

Direct Computation

For v_a, v_b ∈ V:

which, by additivity of ω_1 and ω_2 individually, equals ω_1(v_a) + ω_1(v_b) + ω_2(v_a) + ω_2(v_b), and regrouping gives (ω_1(v_a) + ω_2(v_a)) + (ω_1(v_b) + ω_2(v_b)) = (ω_1 + ω_2)(v_a) + (ω_1 + ω_2)(v_b), exactly the additivity condition required of ω_1 + ω_2.

The Role of Commutativity of Addition in F

The regrouping step relies on commutativity and associativity of addition in the scalar field F, rearranging ω_1(v_a) + ω_1(v_b) + ω_2(v_a) + ω_2(v_b) into the paired form needed; this is a mild but necessary use of the field's own additive structure, underscoring that additivity preservation for functionals rests on the arithmetic of F in addition to the additivity of ω_1 and ω_2 themselves.

Verification of the Homogeneous Condition

Direct Computation

For v ∈ V and a scalar c:

using homogeneity of ω_1 and ω_2 individually, then distributivity of scalar multiplication over addition in F to factor out c. This confirms the second half of the membership test for ω_1 + ω_2.

Closure of V* Under Addition

V* as a Closed Set Under the Pointwise Sum

Because both conditions hold for arbitrary ω_1, ω_2 ∈ V*, the set V* is closed under pointwise addition: the sum of two linear functionals is always another linear functional, never a function lying outside V*. This closure is one of the axioms required for V*, equipped with pointwise addition, to qualify as a vector space in its own right, alongside the corresponding closure under scalar multiplication.

Necessity for the Dual Space to Be Well Defined

Without additivity preservation, the phrase "V* is a vector space" would be unjustified, since the addition operation used to define vector-space structure on V* would risk producing elements falling outside V* itself; additivity preservation is therefore not a supplementary observation but a load-bearing requirement for the basic definition of the dual space to be internally consistent.