12.11.5 Tensor Restriction Resulting Tensor

Tensor Restriction Resulting Tensor is the concrete multilinear object produced once domain restriction has been carried out, characterized as a genuine tensor of the same type as the original but defined entirely over the chosen target subspace rather than over the original, larger source domain.

Nature of the Resulting Object

A Genuine Tensor in Its Own Right

The resulting tensor, often denoted $$A_{|W}$$ for an original tensor $$A$$ restricted to a target subspace $$W$$, is not merely a notational convenience but a fully legitimate multilinear map in its own right, satisfying every property required of a tensor when considered over the smaller vector space $$W$$.

Same Type, Smaller Domain

If the original tensor $$A$$ has type $$(p,q)$$, the resulting restricted tensor retains exactly that same type $$(p,q)$$, differing from the original only in that its underlying vector space is now the target subspace $$W$$ rather than the original source domain $$V$$.

Componentwise Characterization

Inherited Components Restricted to the Subspace

Relative to a basis of $$W$$, extended if needed to a basis of $$V$$, the components of the resulting tensor are precisely those components of the original tensor $$A$$ whose indices range only over the basis vectors belonging to $$W$$:

for index values restricted to those associated with the basis of $$W$$.

Properties Inherited from the Original Tensor

Multilinearity Preserved

The resulting tensor retains full multilinearity with respect to its arguments, since its defining rule is simply the original tensor's multilinear formula, now applied only to arguments drawn from $$W$$, and restriction of a multilinear function's domain to a subspace remains multilinear on that subspace.

Compatible with Further Tensor Operations

Because the resulting tensor is a genuine tensor of type $$(p,q)$$ over the vector space $$W$$, it can be freely combined with other tensors of the same type built over that same $$W$$, using addition, subtraction, scalar multiplication, or further restriction, exactly as any ordinary tensor over $$W$$ would be.

Relationship to the Original Tensor

Loss of Information Outside the Subspace

The resulting tensor no longer carries any information about how the original tensor $$A$$ behaves on vectors or covectors outside the target subspace $$W$$, since those directions have been entirely excluded from consideration during the restriction.

Full Recoverability Is Not Guaranteed

Because the restricted tensor discards this outside information, it is generally not possible to reconstruct the original tensor $$A$$ in full from the resulting restricted tensor alone, unless the target subspace happened to coincide with the entire original source domain.

Illustration