12.12.3 Tensor Extension Compatibility Requirement

Tensor Extension Compatibility Requirement is the condition that must hold before domain extension can be meaningfully performed, requiring that the larger target vector space genuinely contain the original tensor's domain as a subspace and that any chosen extension rule agree exactly with the original tensor wherever the original domain applies.

Components of the Compatibility Requirement

Genuine Containment of the Original Domain

For a tensor $$A$$ defined over a vector space $$W$$, extension to a larger space $$V$$ requires that $$W$$ genuinely be a subspace of $$V$$, satisfying $$W\subseteq V$$ in the strict sense of vector subspace containment, not merely having the same dimension or being isomorphic to a subspace of $$V$$.

Agreement of the Extension Rule on the Original Domain

Beyond containment, the extended tensor $$A^{~}$$ must satisfy:

for every choice of arguments drawn entirely from $$W$$, ensuring the extension does not silently alter the values the original tensor already assigned on its native domain.

Why This Requirement Is Necessary

Preventing an Ill-Defined or Misleading Extension

Without genuine containment of $$W$$ within $$V$$, there would be no meaningful sense in which the extended tensor's domain actually includes the original tensor's domain, undermining the basic purpose of extension, which is to broaden, not replace, the tensor's applicability.

Preserving the Identity of the Original Tensor

Requiring agreement on the original domain ensures that the extended tensor can genuinely be regarded as an extension of $$A$$, rather than an unrelated tensor that merely happens to be defined on a superset of $$W$$. Without this agreement, calling the result an extension of $$A$$ would be misleading.

Consequences for the Extension Rule

Freedom Constrained Only Outside the Original Domain

The compatibility requirement constrains the extension rule only insofar as it must reproduce the original tensor's values on $$W$$; it leaves the extension rule free to be specified in whatever manner is convenient or appropriate for arguments involving directions outside $$W$$, such as the common convention of extension by zero.

Verifying Compatibility Before Relying on an Extension

Before using a proposed extended tensor in further work, it is necessary to confirm that both aspects of the compatibility requirement hold: that the claimed containment of $$W$$ within $$V$$ is genuine, and that the extension's values agree with the original tensor on that shared domain.

Relationship to Restriction

Complementary Compatibility Between the Two Operations

This compatibility requirement mirrors the evaluation preservation property of domain restriction, since restricting a compatible extension back to its original domain $$W$$ must recover the original tensor exactly, confirming that extension and restriction, when performed compatibly with one another, undo each other's effect on the shared subspace.

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