14.7.4 Tensor Map Product Domain Compatibility

Tensor Map Product Domain Compatibility is the requirement that the stated domains of the two maps forming a factor map pair match, factor by factor, the actual spaces used to build the domain tensor product, without which the tensor product of maps cannot be regarded as acting on that particular tensor product at all.

Statement of the Compatibility Requirement

Matching Domains to Tensor Factors

For a tensor product of maps to act on a specific domain tensor space V1 tensor V2, domain compatibility requires that the map f assigned to the first factor have domain exactly V1, and the map g assigned to the second factor have domain exactly V2,

Without this exact match, the elementary output rule f(v) tensor g(w) cannot even be evaluated, since f would not accept v as an input, or g would not accept w as an input, when the stated domains fail to coincide with the intended tensor factors.

Compatibility Is Factorwise, Not Aggregate

Domain compatibility must hold on each factor individually rather than merely requiring the two domains to have compatible total dimension or some other aggregate property, since a mismatch on even one factor, such as f having domain V1 prime instead of V1, prevents the construction from applying to the intended tensor product, regardless of any coincidental agreement in dimension between V1 prime and V1.

Consequences of Domain Compatibility

Necessity for the Elementary Action to Be Defined

Domain compatibility is what permits the elementary action

to be evaluated in the first place, since evaluating f at v presupposes that v lies in the domain of f, a presupposition guaranteed only when domain compatibility holds between f and the first domain factor space.

Failure of Compatibility Blocks the Construction Entirely

If domain compatibility fails, no amount of adjustment to the codomain side of the construction can repair the tensor product of maps, since the failure occurs before any output is produced; restoring domain compatibility requires either replacing the map with one whose domain matches the intended factor, or replacing the intended tensor factor with the actual domain of the given map.

Domain Compatibility Under Restriction

Restoring Compatibility by Restriction

If a map f has domain strictly larger than the intended tensor factor V1, domain compatibility can be restored by restricting f to V1, producing a new map whose domain now matches exactly, provided V1 is indeed a subspace of the original domain of f; this restricted map, rather than the original f, is the one entering into a domain-compatible tensor product of maps.

Compatibility Preserved Under Passing to Invariant Subspaces

When forming the tensor product of maps restricted to invariant subspaces U of V1 and X of V2, domain compatibility is preserved automatically, since the restricted maps have domains exactly U and X, matching the tensor factors U and X of the smaller domain tensor space U tensor X under consideration.

Domain Compatibility Under Composition

Compatibility Required at Every Stage of a Composition Chain

When composing several tensor products of maps in sequence, domain compatibility must hold not only for the first tensor product of maps in the chain, matching its domain factor spaces to the overall starting tensor product, but also implicitly for every subsequent stage, where the codomain factor spaces of one stage must serve correctly as the domain factor spaces of the next.

Compatibility as a Precondition for the Composition Identity

The composition identity for tensor products of maps presupposes domain compatibility throughout the chain being composed, since the identity itself is stated for maps already known to act correctly between the relevant domain and codomain tensor spaces; domain compatibility is therefore a standing precondition for the composition identity to even be a meaningful statement, rather than a consequence derived from it.