5.21.5 Tensor Product Higher Product Preparation

Tensor Product Higher Product Preparation is the body of associativity results that establish, in advance of defining an n-factor tensor product directly, that any iterated construction built from the ordinary pairwise (two-factor) tensor product produces a space canonically isomorphic to the one obtained from a direct multilinear universal property, so that the two apparently different routes to a tensor product of many factors may be used interchangeably. This preparation is what licenses treating iterated binary tensoring and genuinely n-ary tensor products as the same object, rather than requiring a separate, independent construction and theory for each number of factors.

Two Routes to a Multi-Factor Tensor Product

Route One: Iterated Binary Tensoring

Starting only from the two-factor tensor product, a product of n spaces can be built by repeatedly applying ⊗ two spaces at a time, for instance as (((V₁ ⊗ V₂) ⊗ V₃) ⊗ ... ) ⊗ Vₙ, using only the original pairwise construction and its universal property at each step.

Route Two: Direct Multilinear Universal Property

Alternatively, V₁ ⊗ V₂ ⊗ ... ⊗ Vₙ can be defined directly as the space satisfying a universal property for n-linear (not merely bilinear) maps: for every n-linear map μ : V₁ × V₂ × ... × Vₙ → U, there is a unique linear map μ̃ from the n-factor tensor product to U satisfying μ̃(v₁ ⊗ ... ⊗ vₙ) = μ(v₁, ..., vₙ).

Why the Two Routes Must Be Reconciled

The Preparation Problem

Without further argument, it is not obvious that the iterated binary construction of Route One satisfies the same universal property demanded of Route Two, since Route One is defined by repeated application of the two-factor universal property, while Route Two's defining property speaks about n-linear maps directly, a structurally different condition.

Associativity as the Bridge

Higher product preparation solves this by showing that any two iterated binary parenthesizations of V₁, ..., Vₙ are canonically isomorphic (associativity), and further, that the resulting space — however parenthesized — does satisfy the n-linear universal property of Route Two. This second fact is proved by an inductive argument: an n-linear map corresponds to a family of (n-1)-linear maps depending linearly on one extra variable, which by induction factor through iterated applications of the two-factor universal property.

Diagram of the Reconciliation

Inductive Verification of the n-Linear Property

Base Case

For n = 2, the claim is exactly the defining universal property of the ordinary tensor product, requiring no further argument.

Inductive Step

Assuming the (n-1)-factor iterated construction satisfies the (n-1)-linear universal property, an n-linear map μ on V₁ × ... × Vₙ can be regarded, for each fixed vₙ, as an (n-1)-linear map in the remaining variables, inducing (by the inductive hypothesis) a linear map on V₁ ⊗ ... ⊗ V_{n-1} depending linearly on vₙ; this dependence is itself bilinear jointly, so applying the two-factor universal property once more produces the required unique linear map on (V₁ ⊗ ... ⊗ V_{n-1}) ⊗ Vₙ, completing the induction.

Consequences for Working with Higher Tensor Products

Freedom to Choose the More Convenient Definition

Once higher product preparation is established, one may freely switch between thinking of V₁ ⊗ ... ⊗ Vₙ as built from iterated pairwise tensoring (useful for constructive, step-by-step arguments) or as the object satisfying the direct n-linear universal property (useful for defining maps out of the space in one step from an n-linear map), knowing both descriptions agree.

Basis and Dimension Consistency

The preparation also confirms that the basis {eᵢ₁ ⊗ eᵢ₂ ⊗ ... ⊗ eᵢₙ} obtained from bases of the individual factors, however it is parenthesized during construction, gives a single well-defined basis of the resulting space with dimension equal to the product dim(V₁) · dim(V₂) · ... · dim(Vₙ), regardless of which route was used to build the tensor product.

Significance of Higher Product Preparation

Groundwork for Tensor Algebra Constructions

Higher product preparation is the technical groundwork required before defining tensor powers V^{⊗n}, the full tensor algebra T(V) = ⊕ₙ V^{⊗n}, and related constructions such as symmetric and exterior powers, all of which rely on being able to treat an n-fold tensor product as a single coherent object rather than a fragile iterated construction.

Avoiding Redundant Redefinition

By establishing once and for all that iterated pairwise tensoring agrees with the direct multilinear definition, higher product preparation avoids the need to redefine, and re-verify a universal property for, tensor products of each new number of factors from scratch, allowing all higher-order tensor theory to build directly on the ordinary two-factor tensor product and its associativity.