12.11.3 Tensor Restriction Slot Limitation

Tensor Restriction Slot Limitation is the constraint imposed by domain restriction whereby every one of a tensor's argument slots, not merely some of them, is confined to accept only vectors and covectors drawn from the chosen subspace, so that no slot retains the freedom to accept an argument from outside that subspace.

The Nature of the Limitation

Every Slot Is Constrained Simultaneously

When a tensor $$A$$ of type $$(p,q)$$ undergoes domain restriction to a subspace $$W\subseteq V$$, the limitation applies uniformly across all $$p$$ covector slots and all $$q$$ vector slots at once. It is not possible to restrict only a subset of the tensor's slots while leaving the remaining slots open to the full original space; the restriction is total across every slot.

Formal Statement of the Limitation

For the domain-restricted tensor $$A_{|W}$$, every argument supplied to any slot must satisfy:

for each of the $$q$$ vector slots, and an analogous condition using the appropriately restricted dual space for each of the $$p$$ covector slots, with no slot exempted from this requirement.

Why the Limitation Applies Uniformly

Consistency of the Multilinear Structure

Because a tensor's multilinear structure treats its slots jointly rather than independently, allowing some slots to remain unrestricted while others are confined to $$W$$ would produce an object whose domain of definition is not a clean, well-characterized subspace, undermining the very purpose of performing a restriction in the first place.

Alignment with the Restricted Vector Space

Since the entire purpose of restriction is to reinterpret the tensor as acting over the smaller space $$W$$ rather than the original $$V$$, every slot must correspondingly be understood as accepting arguments from this smaller space, since the tensor's new underlying vector space is $$W$$ in its entirety, not a mixture of $$W$$ in some slots and $$V$$ in others.

Consequences of the Slot Limitation

Arguments Outside the Subspace Are Rejected

Any attempt to supply a vector or covector that lies outside the chosen subspace $$W$$ to any slot of the restricted tensor falls outside the domain on which the restricted tensor is defined, regardless of which particular slot the outside argument was intended for.

Full Consistency Across the Restricted Tensor

Because the limitation is uniform, the restricted tensor behaves exactly like an ordinary tensor over the smaller vector space $$W$$, with no lingering trace of the original larger space $$V$$ visible in any individual slot's behavior.

Relationship to the Overall Restriction Operation

Slot Limitation as the Defining Feature

While the domain restriction operation as a whole describes reinterpreting a tensor over a smaller vector space, slot limitation specifically emphasizes that this reinterpretation touches every slot of the tensor without exception, distinguishing a genuine domain restriction from any partial or selective narrowing of only some arguments.

Illustration