2.22.5 Tensor Linear Map Tensorial Preparation

Tensor Linear Map Tensorial Preparation is the preliminary bilinearity argument that must be established before the identification V* ⊗ W ≅ Hom(V, W) can be constructed by the universal property of the tensor product, consisting of the verification that the assignment sending a pair (ω, w) ∈ V* × W to the rank-one linear map v ↦ ω(v) w is bilinear in ω and w separately. This bilinearity is exactly the hypothesis the universal property of the tensor product requires in order to guarantee a unique linear map out of V* ⊗ W factoring through the pairing, and without first confirming it, the identification of linear maps with tensors would have no foundation.

The Universal Property Requirement

What the Universal Property Demands

The defining universal property of V* ⊗ W states that for any vector space X and any bilinear map β : V* × W → X, there exists a unique linear map β̄ : V* ⊗ W → X satisfying β̄(ω ⊗ w) = β(ω, w) for all ω ∈ V* and w ∈ W. To apply this property in order to build the identification with Hom(V, W), the candidate map β : V* × W → Hom(V, W), defined by β(ω, w) = (v ↦ ω(v) w), must first be shown to be bilinear; tensorial preparation is precisely this verification step.

Why Bilinearity Cannot Be Assumed

The pairing β(ω, w) = (v ↦ ω(v) w) is a specific construction, not an abstract given, so its bilinearity is a fact that must be checked directly from the definitions of vector addition and scalar multiplication in Hom(V, W), in V*, and in W, rather than something automatically true simply because the tensor product is involved.

Verifying Linearity in the First Argument

Additivity in ω

For ω_1, ω_2 ∈ V* and w ∈ W, additivity of β in its first argument requires β(ω_1 + ω_2, w) = β(ω_1, w) + β(ω_2, w) as elements of Hom(V, W). Evaluating both sides at an arbitrary v ∈ V:

which holds because (ω_1 + ω_2)(v) = ω_1(v) + ω_2(v) by the definition of addition in V*, and scalar multiplication in W distributes over this sum.

Homogeneity in ω

For a scalar c, β(cω, w) = (v ↦ (cω)(v) w) = (v ↦ c(ω(v)) w) = c · β(ω, w), using (cω)(v) = c(ω(v)), the definition of scalar multiplication in V*. Together with additivity, this confirms β is linear in its first argument.

Verifying Linearity in the Second Argument

Additivity and Homogeneity in w

For w_1, w_2 ∈ W, β(ω, w_1 + w_2) = (v ↦ ω(v)(w_1 + w_2)) = (v ↦ ω(v) w_1 + ω(v) w_2) = β(ω, w_1) + β(ω, w_2), using distributivity of scalar multiplication over vector addition in W. For a scalar c, β(ω, cw) = (v ↦ ω(v)(cw)) = (v ↦ c(ω(v) w)) = c · β(ω, w), using the compatibility of scalar multiplication with the field in W. These two facts together establish linearity of β in its second argument.

Bilinearity as the Conjunction of Both

Because β is linear separately in ω, with w held fixed, and separately in w, with ω held fixed, β is bilinear in the technical sense required by the universal property, which is a strictly weaker condition than joint linearity in the pair (ω, w) treated as a single vector in V* ⊕ W; bilinearity concerns each argument's behavior individually, not the combined behavior of both arguments changing simultaneously.

Consequence: The Identification Map Exists and Is Well Defined

Applying the Universal Property

With bilinearity of β established, the universal property of the tensor product guarantees a unique linear map Θ = β̄ : V* ⊗ W → Hom(V, W) satisfying Θ(ω ⊗ w) = β(ω, w) on simple tensors, and this Θ is exactly the identification map used throughout tensor linear map structure, additivity preservation, and scalar preservation. Tensorial preparation is the specific bilinearity check that makes the existence and uniqueness of Θ a consequence of the universal property rather than an unjustified assumption.

Downstream Reliance on This Preparation

Every later claim about Θ, that it preserves sums, that it preserves scalar multiples, that it is bijective and hence an isomorphism, presupposes that Θ is well defined in the first place, and it is tensorial preparation, the verification of bilinearity of β, that supplies this prerequisite.