6.6.2 Tensor Arity Multilinear Map Relation

Tensor Arity Multilinear Map Relation is the precise statement, grounded in the universal property of the tensor product, that a tensor viewed as an abstract element of V₁ ⊗ ... ⊗ V_k's dual (or, dually, of the tensor product of duals) corresponds bijectively to a k-linear map V₁ × ... × V_k → F, and that this correspondence is exactly what licenses calling a tensor "k-ary" in the first place. Without this relation, arity classification would have no justified connection between the algebraic object (an element of a tensor product space) and the functional description (a multilinear map with k inputs) that arity terminology is built to describe.

Stating the Correspondence

The Universal Property as a Bijection

The universal property of the tensor product asserts a natural bijection:

between linear functionals on the tensor product and multilinear maps on the original factors, with the arity k appearing on both sides as, respectively, the number of tensor-product factors and the number of multilinear arguments.

Direction One: From a Linear Functional to a Multilinear Map

Given a linear functional φ on V₁ ⊗ ... ⊗ V_k, define a k-linear map by f(v₁, ..., v_k) = φ(v₁ ⊗ ... ⊗ v_k); multilinearity of f follows immediately from linearity of φ together with the multilinearity of the tensor product map itself.

Direction Two: From a Multilinear Map Back to a Linear Functional

Given a k-linear map f, the universal property guarantees a unique linear functional φ on V₁ ⊗ ... ⊗ V_k satisfying φ(v₁ ⊗ ... ⊗ v_k) = f(v₁, ..., v_k) on every simple tensor, and this assignment extends uniquely, by linearity, to all of V₁ ⊗ ... ⊗ V_k, since simple tensors span the whole space.

Diagram of the Two-Way Correspondence

Why the Correspondence Is Exactly Bijective

Simple Tensors Span, So No Information Is Lost Going One Way

Because simple tensors v₁ ⊗ ... ⊗ v_k span the entire tensor product space, a linear functional is completely determined by its values on simple tensors alone, so the restriction map from functionals to multilinear maps loses no information: two different functionals cannot restrict to the same multilinear map.

The Universal Property Guarantees Existence Going the Other Way

The existence half of the universal property guarantees that every k-linear map does arise from some functional on the tensor product, so the correspondence is not merely injective but also surjective, giving the full bijection stated above.

Arity as the Invariant Preserved by the Correspondence

The Number k Appears Identically on Both Sides

Throughout the correspondence, the number k never changes: a functional on a k-factor tensor product corresponds to a map with exactly k arguments, never more and never fewer; arity is precisely the quantity this bijection carries across from the algebraic to the functional description without alteration.

Why This Justifies Arity Terminology for Tensors

Calling an element of V₁ ⊗ ... ⊗ V_k's dual a "k-ary tensor" is only justified because the multilinear-map relation guarantees that such an element genuinely does correspond to, and behave exactly like, a k-argument function; without this relation, arity classification would be borrowing function-theoretic language without any guarantee that tensors actually possess the arity-like behavior that language implies.

Consequences for Arity-Based Operations

Tensor Product of Functionals Corresponds to Concatenated Multilinear Maps

Under the correspondence, the tensor product of two functionals (an operation with arity k₁ + k₂ on the algebraic side) corresponds exactly to combining a k₁-linear map and a k₂-linear map into a single (k₁+k₂)-linear map on the functional side, confirming that arity addition under the tensor product is not merely an analogous rule on two sides but the same operation viewed through two lenses.

Contraction Corresponds to Partial Evaluation Against a Pairing

Contraction on the algebraic side corresponds, under the relation, to composing the multilinear map with a canonical evaluation pairing on two of its arguments, reducing arity by 2 on the functional side exactly as it does on the algebraic side, again confirming the two descriptions are two views of a single underlying operation.

Why This Relation Matters

The Foundation Beneath All of Arity Classification

Every other fact stated under arity classification — the naming ladder nullary through k-ary, currying, composition — presupposes that tensors genuinely are k-argument functions in a precise sense, and this relation is the specific theorem that supplies that precision, rather than leaving it as an informal analogy.

Enabling Free Movement Between Algebraic and Functional Reasoning

Because the correspondence is an exact bijection respecting all the relevant operations, a proof or calculation can move freely between treating a tensor as an abstract algebraic element and treating it as a concrete multilinear function, using whichever framing is more convenient at each step, with the arity multilinear map relation guaranteeing the two framings never disagree.