14.3.2 Tensor Map Product Factor Space Relation

The Tensor Map Product Factor Space Relation explores how tensor maps interact with factor spaces, revealing structural connections in algebraic tensor algebra frameworks.

Tensor Map Product Factor Space Relation is the correspondence linking the domain and codomain spaces of each map in a factor map pair to the two tensor factors that appear in the domain and codomain of the resulting tensor product map, fixing exactly which ambient tensor products the construction connects.

The Basic Correspondence

Domains Correspond to Domain Factors

For a factor map pair consisting of

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

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

the factor space relation identifies V1 as the first factor of the domain of f tensor g and V2 as the second factor, so the domain of the tensor product map is precisely

V_{1} \otimes V_{2} .

Codomains Correspond to Codomain Factors

Symmetrically, the factor space relation identifies W1 as the first factor of the codomain of f tensor g and W2 as the second factor, so the codomain of the tensor product map is precisely

W_{1} \otimes W_{2} .

Together, these two correspondences fully determine the pair of tensor products connected by f tensor g directly from the individual domains and codomains of f and g, without any further specification being required.

Consistency of the Relation Under Operations

Preservation Under Composition

When two tensor products of maps are composed, the factor space relation for the composite is obtained by combining the relations for the two pieces: if f tensor g maps V1 tensor V2 into W1 tensor W2, and f prime tensor g prime maps W1 tensor W2 into U1 tensor U2, then the factor space relation for the composite

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

identifies V1 tensor V2 as the domain and U1 tensor U2 as the codomain, matching exactly the domain and codomain that would be assigned to the single tensor product of composites

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

This agreement is required for the composition identity of tensor products of maps to be well posed in the first place, since both sides of that identity must describe maps between the same pair of spaces.

Preservation Under Identity Extension

Extending a single map f by an identity map on an auxiliary space W preserves the factor space relation on the factor carrying f, while introducing W itself as the factor space relation on the new factor, so that f tensor identity-on-W relates V to W1 on the first factor and relates W to itself on the second factor, matching the general pattern exactly.

Constraints Imposed by the Relation

Matching Requirement for Well-Defined Products

The factor space relation imposes a strict matching requirement: a tensor product of maps f tensor g can only be regarded as a map on a specific tensor product V1 tensor V2 if the stated domains of f and g coincide, factor by factor, with V1 and V2 respectively. If a different pairing of tensor factors is intended, the underlying maps in the factor map pair must be reindexed or reordered to match, since the factor space relation does not permit any implicit reinterpretation of which map corresponds to which factor.

Consequences for Substitution

Because the factor space relation ties each map rigidly to its designated factor, substituting a map with a different domain or codomain into a factor map pair changes the factor space relation of the resulting tensor product map accordingly, and any construction relying on the original factor spaces, such as a chosen basis or a fixed subspace, must be reconsidered relative to the new domains and codomains introduced by the substitution.