2.19.5 Tensor Abstract Tensor Construction

Tensor Abstract Tensor Construction is the explicit procedure by which a tensor product space, and more generally any type (p, q) tensor space, is built out of abstract vector spaces, together with the proof that the resulting space satisfies the universal property used to characterize it. Where the abstract vector space context establishes what properties a tensor space must have, and abstract element and operation handling describe how to work with the objects and maps once they exist, abstract tensor construction supplies the actual manufacturing process, most commonly the free vector space quotient construction, that guarantees such a space exists at all and is unique up to a canonical isomorphism.

The Free Vector Space Quotient Construction

Step One: The Free Vector Space on Pairs

Given abstract vector spaces V and W over a field F, the construction begins with the free vector space F(V × W) generated by the set V × W, meaning the vector space whose elements are finite formal linear combinations of ordered pairs (v, w), with no relations imposed between different pairs at this stage; every pair is treated as an independent basis vector of this (typically infinite-dimensional) space.

Step Two: Quotienting by the Bilinearity Relations

A subspace R of F(V × W) is generated by all elements of the form

together with the analogous relation in the second argument and the scalar compatibility relation (α v, w) - α(v, w). The tensor product is then defined as the quotient V ⊗ W := F(V × W) / R, and the image of (v, w) in this quotient is written v ⊗ w. Passing to the quotient is exactly what forces the bilinearity relations to hold, since any two formal sums differing by an element of R are identified in V ⊗ W.

Verifying the Universal Property Holds

To confirm the construction is correct, one checks that the canonical map ⊗ : V × W → V ⊗ W is bilinear by construction, and that every bilinear map B : V × W → U factors uniquely through it: B extends to a linear map on the free vector space F(V × W) by linearity on the generating pairs, this extension vanishes on R precisely because B is bilinear, and it therefore descends to a well-defined linear map on the quotient V ⊗ W, which is the required unique factorization.

Uniqueness Up to Canonical Isomorphism

Why the Construction's Details Do Not Matter Afterward

Any two vector spaces satisfying the universal property of the tensor product for the same V and W are related by a unique isomorphism compatible with the canonical bilinear maps into each. This is a standard consequence of the universal property itself, applied twice in opposite directions, and it means that once the free vector space quotient construction has verified existence, the specific formal machinery used, formal sums, equivalence classes, generating relations, can be forgotten, and V ⊗ W can be treated purely through its universal property from then on.

Extending the Construction to Type (p, q) Tensor Spaces

Iterating the Tensor Product

The construction extends from a single tensor product of two spaces to a full type (p, q) tensor space by iterating it: V^{⊗p} ⊗ (V^{*})^{⊗q} is built by repeatedly applying the two-factor construction, using the associativity isomorphism (A ⊗ B) ⊗ C ⊥ A ⊗ (B ⊗ C), itself proved from the universal property, to justify treating the result as a single, well-defined p + q-fold tensor product independent of the order in which the factors are combined.

Constructing the Dual Space Factor

The dual space V* used as a factor in this iterated construction is itself built abstractly, as the vector space of all linear functionals V → F, before any tensor product is taken; its inclusion as a factor is what allows the same free-vector-space-quotient method to produce mixed tensors carrying both contravariant and covariant slots, rather than only tensor powers of V alone.

An Alternative Construction via Multilinear Functionals

Tensors as Functionals on Multilinear Maps

A second construction defines V ⊗ W, for finite-dimensional V and W, as the space of bilinear functionals on V* × W*, using the natural identification of a finite-dimensional space with its double dual. This construction produces a space isomorphic to the quotient construction, illustrating that the abstract tensor space is genuinely characterized by its universal property alone, since two structurally different construction methods yield canonically isomorphic results.

Diagrammatic Summary

The diagram traces the construction from the free vector space on pairs (v, w), through the quotient by the bilinearity relations R, to the resulting tensor product space V ⊗ W, the abstract object whose existence and uniqueness this construction procedure establishes.