14.1.2 Tensor Map Product Domain Scope

The Tensor Map Product Domain Scope defines how tensor maps interact within their algebraic structure, establishing boundaries for their operations and transformations.

Tensor Map Product Domain Scope is the precise specification of which domains a tensor product of linear maps is permitted to act on, fixing the pair of source spaces that the construction requires before the induced map on the tensor product can be formed at all.

Purpose of the Domain Scope

Why a Scope Is Needed

The tensor product of two linear maps f and g is only defined once the domain of f and the domain of g are separately fixed as the two tensor factors of a single product space. The domain scope is the bookkeeping device that records this pairing: it declares that f acts on the first factor and g acts on the second factor, and it forbids any silent reordering or merging of factors that would change which map acts on which component.

Distinguishing Domain Scope from Codomain

The domain scope concerns only the source side of the construction. Given

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

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

the domain scope is the ordered pair of source spaces

(V_{1}, V_{2})

together with the resulting product domain

V_{1} \otimes V_{2} .

The target spaces and their ordering play no role in fixing the domain scope, since the same domain scope may be paired with different codomains by choosing different maps out of the same sources.

Ordering and Positional Assignment

Position Determines Action

Within the domain scope, position is meaningful: the map occupying the first slot always acts on the first tensor factor, and the map occupying the second slot always acts on the second tensor factor. Swapping the two maps without also swapping the two factors changes the resulting map on the tensor product, in general producing a different linear map even when the two factors are isomorphic spaces.

Scope Under Permutation of Factors

If a permutation is applied to the tensor factors of the domain, exchanging the roles of the first and second factor, the domain scope of the maps acting on the reordered product must be permuted identically for the induced map to remain well defined. This requirement keeps the assignment of each map to its intended factor consistent under any relabeling of the underlying tensor product.

Compatibility Requirements Within the Scope

Matching of Source Spaces

A tensor product of maps can be formed only when the declared domain scope matches, factor by factor, the actual vector spaces being tensored. If the first factor of the tensor product is a space distinct from the stated domain of the first map, the construction is not applicable, and no induced map on the tensor product exists until the mismatch is corrected, either by restricting the domain of the map or by choosing a different tensor product whose first factor agrees with it.

Scope Restriction to Subspaces

When a map is defined on a domain that properly contains one of the tensor factors, the domain scope for the purposes of the tensor product construction is understood as the restriction of that map to the relevant factor. This restricted map, rather than the original map on its larger domain, is the one entering into the tensor product, and its domain scope is the tensor factor itself rather than the ambient space.

Consequences for Composability

Sequential Domain Scope Matching

When composing a tensor product of maps with another tensor product of maps, the domain scope of the second construction must match the codomain scope of the first, factor by factor. This factorwise matching is what permits the composition rule

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

to hold, since each side is only meaningful once the domain scopes of the two tensor products align on each factor separately.

Scope Preservation Under Identity Extension

Extending a single map by the identity on an additional factor keeps the domain scope of the original map unchanged on its own factor while adding the identity's domain as a new factor in the scope. This preserves the original map's behavior on its designated factor exactly, ensuring that identity extension never alters the domain scope already fixed for the map being extended.