12.19.4 Tensor Restriction Closure Context

Tensor Restriction Closure Context is the specific setting — a designated subspace of an ambient tensor space, together with the operations under consideration — within which restricting a tensor or a tensor operation down to that subspace is guaranteed to produce a result still belonging to the subspace itself, as opposed to a result that only makes sense back in the larger ambient space. Where composition closure context asks whether chaining operations stays inside a designated family of maps, restriction closure context asks the complementary question about narrowing tensors and operations down to a smaller region of a larger space: whether that narrowing is self-consistent, or whether it inevitably leaks back into the ambient structure.

Restriction of a Tensor to a Subspace

Restricting the Domain of a Covariant Tensor

Given a covariant tensor T on an ambient space V and a subspace U ⊆ V, the restriction T|_U is defined by evaluating T only on tuples of vectors drawn from U. Since U is itself a vector space, and T restricted to accept only arguments from U remains multilinear in those arguments, T|_U is again a genuine covariant tensor, now on U rather than on V; this restriction is exactly the pullback of T along the inclusion ι : U → V.

The Restriction Closure Context for This Case

The context here is "covariant tensors defined on subspaces of a fixed ambient space V," and restriction is closed on this context precisely because the pullback along an inclusion always lands in the space of covariant tensors on the smaller subspace, never producing an object that is neither a tensor on U nor expressible purely in terms of U's own structure.

Restriction of an Operation to a Subspace

When a Projection Restricts to an Idempotent Map on the Subspace

If π is a projection on V and U is exactly the image of π, the restriction of π to U is the identity map on U, which trivially stays within the endomorphisms of U. But if U is instead some arbitrary subspace unrelated to the image or kernel of π, the restriction π|_U need not even map into U at all — a vector in U can project to somewhere entirely outside U, so the restriction closure context fails unless U is chosen compatibly with π's own structure.

Compatibility Condition for Restriction Closure

Restriction of an endomorphism π to a subspace U stays closed on U — meaning π|_U is a genuine endomorphism of U rather than merely a map from U into the larger ambient space — exactly when U is invariant under π, satisfying π(U) ⊆ U. This invariance condition is the precise criterion determining the restriction closure context for endomorphisms, distinguishing subspaces on which restriction is meaningful as a self-contained operation from those on which it is not.

Diagram of Restriction Closure Depending on Invariance

Restriction Closure Context Across Several Operations

Direct Sum Summands as an Always-Closed Restriction Context

Each summand Vₖ of a direct sum V = V₁ ⊕ ... ⊕ Vₙ is automatically invariant under its own associated composite ιₖ ∘ πₖ, so restriction of this particular composite operation to Vₖ is always closed by construction, without requiring any additional invariance check beyond what the direct sum's own defining structure already guarantees.

Bilinear Forms Restricted to a Subspace

A bilinear form β on V, restricted to arguments from a subspace U, always produces a bilinear form on U, since bilinearity in each argument is inherited automatically once both arguments are drawn from U. Restriction closure for bilinear (and more generally covariant multilinear) forms therefore holds unconditionally for any subspace U, in sharp contrast to the invariance condition required for restricting an endomorphism.

Consequences of Restriction Closure Failing

Ill-Defined Self-Contained Computation

If restriction is not closed on a chosen subspace U for a given operation, any attempt to treat computations on U as self-contained — ignoring the rest of the ambient space V — will eventually produce a value that must be reinterpreted back in V, since the operation's output is not guaranteed to remain describable using only U's own structure.

Restriction Closure as a Prerequisite for Studying a Subsystem in Isolation

Establishing restriction closure for an operation on a subspace is exactly what justifies studying that subspace as an independent algebraic system in its own right — for instance, treating an invariant subspace of a linear endomorphism as defining its own smaller eigenvalue problem — since without this closure, the subsystem's behavior cannot be fully understood without constant reference back to the larger ambient space it sits inside.