4.8 Tensor Multilinear Codomain Structure

Tensor Multilinear Codomain Structure is the organization of the output side of a tensor's defining map: the specification of the target space that receives the result of evaluating the map on a filled tuple of arguments, together with whatever basis, dimension, and algebraic structure that target space carries. It is the counterpart to the multilinear domain structure, and together the two structures completely specify a multilinear map: the domain structure fixes what goes in, and the codomain structure fixes what kind of object comes out.

Formal Definition

The Codomain as Part of the Map's Signature

For a multilinear map

the codomain structure is the specification of $W$: whether it is the base field $F$ itself, another vector space entirely, or a more elaborate algebraic object such as a space of tensors of some other type. Unlike the domain, which is decomposed into an ordered sequence of factor slots, the codomain is typically a single, unstructured target unless it happens to itself carry additional tensorial structure.

Two Principal Cases

The codomain structure divides into two principal cases: the scalar codomain case, where $W = F$, and the general vector-valued codomain case, where $W$ is a vector space of dimension one or greater. The scalar case is by far the most common in elementary tensor algebra, corresponding to the ordinary type $(r,s)$ tensor, while the vector-valued case arises for tensor-valued maps such as the torsion or curvature operator viewed as producing a vector rather than a number.

Structural Role of the Codomain

Determining the Output's Algebraic Character

The codomain structure fixes what algebraic operations are available on the tensor's output: if $W = F$, outputs can be added and multiplied using field arithmetic; if $W$ is a vector space, outputs can be added and scaled but not, without further structure, multiplied together. The codomain therefore constrains not just what type of object results from evaluation but what can subsequently be done with that result.

Basis and Component Structure on the Output Side

If $W$ has a chosen basis ${f_1, \ldots, f_m}$, evaluating $T$ on any tuple produces an output that can itself be expanded in that basis, yielding $m$ separate scalar-valued component functions $T_1, \ldots, T_m$, each of which is itself a scalar-valued multilinear map on the same domain. The codomain's basis is what allows a vector-valued tensor to be decomposed into a finite collection of ordinary, scalar-valued tensors.

Interaction With the Domain Structure

Codomain Choice and the Universal Property

The universal property of the tensor product is stated relative to a chosen codomain: for any vector space $W$, multilinear maps out of a fixed domain and into $W$ correspond bijectively to linear maps out of the tensor product space and into that same $W$. Varying the codomain structure, while holding the domain fixed, is precisely what generates the correspondence between multilinear maps of different target types and linear maps on the tensor product space.

Composability of Codomain and Domain

When the codomain of one multilinear map coincides with (or maps into) a factor of another multilinear map's domain, the two maps can be composed, producing a new multilinear map whose domain matches the first map's domain and whose codomain matches the second map's codomain. This composability is only possible because the codomain structure is specified precisely enough to determine whether such an alignment exists.

Effect of Linear Post-Composition

If $L : W \to W'$ is a linear map, composing gives $L \circ T$, a multilinear map with the same domain but a new codomain structure $W'$; this operation transforms the codomain freely without requiring any change to the domain, illustrating that domain and codomain structures, while both essential to a full multilinear map, vary independently of one another.

Summary of Key Points

• The multilinear codomain structure specifies the target space that receives the output of a tensor's defining map, complementing the domain structure that specifies the inputs.
• The two principal cases are the scalar codomain, where the target is the base field, and the vector-valued codomain, where the target is a higher-dimensional space.
• A chosen basis on the codomain allows any vector-valued tensor to be decomposed into a finite family of scalar-valued component tensors.
• The universal property of the tensor product is stated relative to a variable codomain, generating the correspondence between multilinear maps and linear maps on the tensor product space.
• Post-composition with a linear map changes the codomain structure of a tensor without affecting its domain structure.