4.1.2 Tensor Multilinear Map Codomain Structure

The codomain of a tensor multilinear map is structured as a tensor product space, encoding multilinearity through bilinear extensions and universal properties.

Tensor Multilinear Map Codomain Structure is the description of the target space into which a tensor's associated multilinear map sends its arguments, together with the algebraic properties that this target space must have for the multilinear correspondence between tensors and maps to hold. For tensors defined over a single vector space V with base field ℝ, the codomain is the field ℝ itself, but the same structural analysis extends to vector-valued and tensor-valued multilinear maps whose codomain is a nontrivial vector space.

The Scalar Codomain of Ordinary Tensors

Why the Codomain Is the Base Field

An ordinary type (p, q) tensor on V is identified with a multilinear map

T : \times_{k = 1}^{p} V^{*} \times \times_{l = 1}^{q} V \to ℝ

and the field ℝ is itself a one-dimensional vector space over itself, which is what allows the codomain to inherit a trivial but well-defined vector space structure: addition of scalars and scalar multiplication of scalars. This one-dimensionality is essential, because it makes the set of all such multilinear maps into a vector space isomorphic to the tensor product space of the domain factors.

Vector Space Structure on the Codomain

Because ℝ is a field, the codomain automatically carries addition and multiplication satisfying the field axioms, and these operations extend pointwise to the space of multilinear maps: for two multilinear maps T and S with the same domain, (T + S)(args) = T(args) + S(args), and (cT)(args) = c · T(args) for a scalar c. This pointwise structure is what makes the set of type (p, q) tensors a vector space of dimension n^(p+q).

Generalized Vector-Valued Codomains

Codomain Beyond the Base Field

The multilinear map formalism extends naturally to maps whose codomain is an arbitrary vector space W rather than ℝ:

B : \times_{k = 1}^{p} V \to W

Such maps are still linear in each argument separately, but their values are vectors in W rather than scalars. Tensor-valued multilinear maps, where W is itself a tensor product space, arise directly when describing operations such as the tensor product of two tensors, viewed as a single multilinear map jointly linear in both factors' arguments.

Codomain and the Universal Property

The universal property of the tensor product states that any multilinear map with codomain W factors uniquely through the tensor product space: for B as above there exists a unique linear map

\tilde{B} : V_{1} \otimes \dots \otimes V_{p} \to W

satisfying B = B̃ ∘ ⊗, where ⊗ is the canonical multilinear map into the tensor product. This factorization is precisely what characterizes the tensor product space up to isomorphism, and it depends entirely on the codomain W being treated as an ordinary vector space, not merely a set.

Codomain Compatibility Under Composition

Composing Codomains with Linear Maps

If g: W → U is linear and B has codomain W, the composite g ∘ B is again multilinear, now with codomain U. This closure property is what allows codomain-changing operations, such as applying a linear functional to a vector-valued multilinear map to recover an ordinary scalar-valued one, and it underlies how pullback and pushforward operations interact consistently with tensor-valued expressions.

Codomain and Bilinear Pairings

A particularly important case is a bilinear pairing V × V* → ℝ given by evaluation, (v, ω) ↦ ω(v). Its codomain is the base field, and this specific pairing is the generating example from which the general codomain structure of tensor multilinear maps is built, since every contraction between an upper and a lower index of a tensor is, at the level of the multilinear map, an application of this evaluation pairing to one covector slot and one vector slot simultaneously.

Structural Summary

Rank-Codomain Relationship

For a purely scalar-valued (p, q) tensor, the codomain contributes no dimension to the total object; all of the tensor's structural complexity resides in the domain factors. When the codomain is instead a vector space W of dimension m, the associated space of multilinear maps has dimension m · n^(p+q), reflecting that the codomain acts as a multiplicative factor on the space of possible multilinear maps rather than as an additional domain slot.