5.20.5 Tensor Product Linear Map Extension

Tensor Product Linear Map Extension is the technique of passing from a bilinear assignment on pairs (v, w) ∈ V × W to a genuine linear map on the tensor product V ⊗ W, by invoking the universal property of the tensor product rather than by naively declaring values on simple tensors and hoping consistency holds automatically. This extension technique is the rigorous justification underlying the tensor product of linear maps operation f ⊗ g, and more generally it is the standard method by which any bilinear (or multilinear) rule is converted into a single linear map defined on the whole tensor product space.

Why Extension Requires Justification

The Naive Approach and Its Danger

It is tempting to define a map on V ⊗ W simply by prescribing its value on each simple tensor, v ⊗ w ↦ f(v) ⊗ g(w), and then extending "by linearity" to sums. This is dangerous because a general element of V ⊗ W can be written as a sum of simple tensors in multiple different ways, and an arbitrary rule on simple tensors need not give the same answer for every such representation, in which case no well-defined linear map results at all.

A Cautionary Non-Example

If one instead attempted to define a map by an assignment on simple tensors that is not compatible with bilinearity — for instance, treating v ⊗ w and w ⊗ v (in a symmetric setting) inconsistently, or assigning values that do not respect (v₁ + v₂) ⊗ w = v₁ ⊗ w + v₂ ⊗ w — the resulting "rule" fails to be well-defined on sums, since different decompositions of the same element would yield conflicting outputs.

The Correct Extension Route: Via Bilinear Maps

Starting from a Genuine Bilinear Map

The safe route begins with an actual bilinear map β : V × W → U for some vector space U — bilinear meaning linear in each argument separately. For the tensor product of linear maps, the relevant bilinear map is β(v, w) = f(v) ⊗ g(w), taking values in U = V′ ⊗ W′.

Invoking the Universal Property

The universal property of V ⊗ W states that for every bilinear map β : V × W → U, there exists a unique linear map β̃ : V ⊗ W → U satisfying

Applying this to β(v, w) = f(v) ⊗ g(w) produces the unique linear map β̃ = f ⊗ g : V ⊗ W → V′ ⊗ W′, and this is precisely how the extension of the elementary action to a genuine linear map is justified — bilinearity of β guarantees β̃ exists and is single-valued, resolving the ambiguity that plagued the naive approach.

Diagram of the Extension

Uniqueness of the Extension

Why the Extension Cannot Be Chosen Differently

Because simple tensors span V ⊗ W, any two linear maps agreeing with β on all simple tensors must agree everywhere on V ⊗ W, since a linear map is determined by its values on a spanning set. This is what makes the extension unique, not merely existent, and it is why writing f ⊗ g unambiguously denotes a single specific linear map rather than one of several possible extensions.

Uniqueness Depends on Spanning, Existence Depends on Bilinearity

The two halves of the universal property serve distinct roles in the extension: bilinearity of β is what makes existence of a consistent linear extension possible at all, while the spanning property of simple tensors is what makes that extension unique once it exists — both facts together give the extension theorem its full strength.

Extension to Multilinear Maps

Generalizing Beyond Two Factors

The same extension technique applies verbatim to multilinear maps on several factors: a multilinear map V₁ × V₂ × ... × Vₙ → U extends uniquely to a linear map on V₁ ⊗ V₂ ⊗ ... ⊗ Vₙ, by the universal property of the multi-factor tensor product, generalizing the two-factor extension used to define f ⊗ g to tensor products of any finite order.

Iterated Extension for Tensor Products of Several Maps

For maps f₁, f₂, ..., fₙ on n factors, the map f₁ ⊗ f₂ ⊗ ... ⊗ fₙ is obtained by the same extension principle applied to the multilinear map (v₁, ..., vₙ) ↦ f₁(v₁) ⊗ ... ⊗ fₙ(vₙ), confirming that the extension technique scales to arbitrarily many tensor factors without modification.

Significance of the Extension Technique

The General Method Behind Every Tensor Product Construction

The extension technique — start with a bilinear or multilinear map, invoke the universal property, obtain a unique linear map on the tensor product — is not specific to f ⊗ g; it is the general method by which essentially every linear map defined "on" a tensor product in practice is actually constructed and justified, including trace maps, contraction maps, and evaluation pairings.

Separating Definition from Verification

By routing the definition of f ⊗ g through this extension technique rather than through a direct, ad hoc formula on general elements, the well-definedness of the tensor product of linear maps operation is settled once and for all at the level of the universal property, freeing every subsequent computation with f ⊗ g from having to re-verify consistency across different representations of the same tensor.