12.18.2 Tensor Operation Space Compatibility

Tensor Operation Space Compatibility is the requirement that the specific vector spaces underlying the tensors and auxiliary maps involved in a tensor operation stand in the precise relationship — identity, subspace inclusion, direct-sum membership, or connection via a fixed linear map — that the operation's defining formula presupposes, independent of whether the variance types or base ring already agree. Two tensors can share the same variance type and be defined over the same field while still being space-incompatible for a given operation, if the particular vector spaces they live on are not related in the way that operation specifically requires.

Identity of Ambient Space as the Simplest Requirement

Same-Space Requirement for Ordinary Addition and Contraction

Adding two tensors, or contracting slots of a single tensor built as a tensor product of several factors, requires every tensor involved to be built from powers of one common space V; there is no meaningful way to add a tensor on V to a tensor on an unrelated space U unless some further structure connects the two. This is the most elementary form of space compatibility, requiring outright identity of the ambient space rather than any weaker relationship.

Subspace Relationship Required for Inclusion-Based Operations

The tensor inclusion operation requires its source subspace and its target space to be related by literal subspace containment (after embedding), not merely by isomorphism; two abstractly isomorphic but unrelated spaces U and U′ do not automatically support an inclusion into a common target space T unless a specific embedding map connecting them has been separately supplied.

Connection via a Fixed Map as a Weaker but Necessary Requirement

Source Map Compatibility for Pullback and Pushforward

The pullback and pushforward operations do not require their two spaces V and W to be identical or for one to contain the other; instead, they require a specifically designated linear map f connecting them, and the operation is compatible only with tensors defined on the domain or codomain that f itself fixes. A tensor on a third, unrelated space U, with no map connecting it to either V or W, is space-incompatible with a pullback or pushforward built from f.

Direct Sum Membership for Injection-Projection Compatibility

An injection ιₖ and a projection πⱼ are compatible with each other's retraction and annihilation identities only when both are built from the same underlying direct-sum decomposition of the same ambient space V. Even if Vₖ and Vⱼ happen to be isomorphic vector spaces, injections and projections drawn from unrelated decompositions of unrelated ambient spaces generally satisfy none of the identities expected of a coherent injection-projection pair.

Diagram of Space Compatibility Requirements Across Operations

Consequences of Space Incompatibility

No Well-Defined Formula Exists

When the space relationship required by an operation is absent, the operation's defining formula has no way to be evaluated: contraction has no shared index range to sum over, an inclusion has no target to embed into, and a pullback or pushforward has no designated map along which to transport the tensor. Space incompatibility of this kind is not a matter of the result being wrong; the expression simply fails to denote anything.

Space Compatibility Must Be Checked Independently of Type Compatibility

A tensor of type (1,1) on V and a tensor of type (1,1) on an unrelated space U are fully type-compatible for contraction — both have the required mix of contravariant and covariant slots — yet remain space-incompatible unless V and U are identified or connected by some specified map, illustrating that variance type agreement, addressed by tensor operation type compatibility, and space agreement, addressed here, are independent conditions that must both be verified.

Establishing Space Compatibility Where None Initially Exists

Introducing an Explicit Connecting Map

When two tensors are defined on genuinely different spaces with no a priori relationship, space compatibility for an operation such as pullback or pushforward can be established by explicitly introducing a linear map between the spaces, converting an otherwise incompatible pair into a compatible one relative to that specific map, though the resulting operation is then relative to, and dependent upon, the particular map chosen.

Restricting to a Common Ambient Structure

When two tensors are defined on subspaces of some larger common space, space compatibility for operations requiring shared ambient space can be established by working within that common ambient space directly — using the inclusion operations of each subspace into the shared structure — converting the original spatial mismatch into one where the compatibility requirement of the intended operation is satisfied by construction.