14.1.3 Tensor Map Product Codomain Scope

The Tensor Map Product Codomain Scope defines the target space for tensor products of linear maps, determining the range of resulting operations.

Tensor Map Product Codomain Scope is the precise specification of which target spaces a tensor product of linear maps is permitted to land in, fixing the pair of destination spaces that the induced map on the tensor product is required to produce as its factorwise images.

Purpose of the Codomain Scope

Why a Scope Is Needed on the Target Side

Once linear maps f and g are tensored, the resulting map lands in a single product space rather than in two separate spaces, so the codomain scope is the record of which two spaces that product space is built from and in what order. It declares that the image of f contributes the first factor of the target and the image of g contributes the second factor, preventing any ambiguity about which component of the output corresponds to which original map.

Distinguishing Codomain Scope from Domain Scope

The codomain scope concerns only the target side of the construction. Given

f : V_{1} \to W_{1}

g : V_{2} \to W_{2}

the codomain scope is the ordered pair of target spaces

(W_{1}, W_{2})

together with the resulting product codomain

W_{1} \otimes W_{2} .

The source spaces and their internal structure play no role in fixing the codomain scope, since the same codomain scope may receive the images of different maps sharing the same pair of targets.

Ordering and Positional Assignment

Position Determines Which Image Lands Where

Within the codomain scope, position is meaningful: the first slot always receives the image of the map assigned to the first tensor factor of the domain, and the second slot always receives the image of the map assigned to the second tensor factor. Exchanging the two target spaces without also exchanging the roles of the maps changes the resulting element of the tensor product, since the factorwise image would then be assembled in the wrong order.

Codomain Scope Under Permutation of Factors

If the tensor factors of the target space are permuted, the codomain scope of the maps producing elements of the reordered product must be permuted in the same way for the induced map to remain consistent with the original construction. Any relabeling of the factors of the tensor product must be tracked simultaneously in the codomain scope of every map contributing to that product.

Compatibility Requirements Within the Scope

Matching of Target Spaces

A tensor product of maps produces a well-defined element of a specific tensor product of target spaces only when the declared codomain scope agrees, factor by factor, with the actual space in which that tensor product is taken. If the first factor of the target tensor product is a space distinct from the stated codomain of the first map, the induced map cannot be regarded as landing in that product until the codomain is corrected or reinterpreted through a further linear map connecting the two spaces.

Codomain Scope Enlargement

When the actual image of a map is a proper subspace of its stated codomain, the codomain scope for the purposes of the tensor product construction may still be taken as the larger stated codomain, since the tensor product of maps is defined relative to the codomain as declared, not relative to the image. Enlarging the codomain scope in this way does not change the values produced by the induced map, only the ambient space in which those values are considered to lie.

Consequences for Composability

Sequential Codomain Scope Matching

When a tensor product of maps is followed by a further tensor product of maps, the codomain scope of the first construction must match the domain scope of the second, factor by factor, for the composite to be defined. This alignment is what allows the factorwise composition rule

(f^{'} \otimes g^{'}) \circ (f \otimes g) = (f^{'} \circ f) \otimes (g^{'} \circ g)

to hold, since the codomain scope of the pair f, g must coincide with the domain scope of the pair f prime, g prime on each factor for both sides of the identity to even be well formed.

Codomain Scope Under Dualization

Passing to dual maps reverses the roles of domain and codomain, so the codomain scope of the original maps becomes, after dualization, the domain scope of the dual maps, and conversely. This exchange is consistent on each tensor factor separately, so dualizing a tensor product of maps produces a tensor product of the dual maps whose domain scope is exactly the codomain scope of the maps being dualized.