4.16.3 Tensor Extension Uniqueness

Tensor Extension Uniqueness is the guarantee that whenever a multilinear extension of prescribed data on a spanning set exists at all, it is the only such extension, so that satisfying tensor extension compatibility not only permits an extension to be built but forecloses any possibility of a second, different tensor also matching the same prescribed data. It completes the tensor multilinear extension construction by addressing the second half of the existence-and-uniqueness pair, showing that compatibility is not merely sufficient for some extension to exist but is also enough to pin down that extension completely.

Statement of Uniqueness

The Claim

Let g_1, ..., g_m span a finite-dimensional vector space V, and suppose prescribed data c(r_1, ..., r_{p+q}) satisfies tensor extension compatibility. If T and T' are both type (p, q) tensors on V satisfying T(g_{r_1}, ..., g_{r_{p+q}}) = c(r_1, ..., r_{p+q}) = T'(g_{r_1}, ..., g_{r_{p+q}}) for every combination of generator indices, then T = T' as multilinear maps on all of V and V*.

Reduction to Vanishing on the Generators

As in the proof of tensor map determination by basis values, it suffices to consider the difference D = T - T', a tensor satisfying D(g_{r_1}, ..., g_{r_{p+q}}) = 0 for every combination of generator indices, and to show that D vanishes on every argument whatsoever.

Proof of Uniqueness

Expanding Arbitrary Arguments in the Generators

Because g_1, ..., g_m spans V, every vector or covector admits at least one expansion as a linear combination of the generators (or their dual analogues). Take an arbitrary input tuple v_1, ..., v_{p+q} and fix one such expansion for each v_l.

Distributing D Across the Expansions

Applying additivity and homogeneity of D in each slot, exactly as in the finite-dimensional basis case, distributes D across every term of every expansion, producing a finite sum in which each term is D evaluated on some combination of the generators g_{r_1}, ..., g_{r_{p+q}}, multiplied by the corresponding expansion coefficients.

Concluding the Proof

Every one of these terms vanishes by hypothesis, since D vanishes on every combination of generators; the entire finite sum is therefore zero, giving D(v_1, ..., v_{p+q}) = 0. Since the input tuple was arbitrary, D vanishes identically, so T = T'.

Uniqueness Does Not Require the Generating Set to Be a Basis

The Proof Uses Only Spanning, Not Independence

Notably, the proof of uniqueness relies only on the fact that g_1, ..., g_m spans V, so that every argument admits at least one expansion in terms of the generators; it does not use linear independence of the generators anywhere, since even if a vector admits several different expansions, the argument only requires the existence of one expansion to carry through.

Why Compatibility Was Still Needed for Existence

Although uniqueness holds regardless of whether the generators are independent, tensor extension compatibility remains necessary to guarantee that an extension exists in the first place when the generators are dependent, since without compatibility a vector's several different expansions could force conflicting values on D, an inconsistency that would prevent any extension, let alone a unique one, from being defined.

Uniqueness and the Basis Case Revisited

Automatic Uniqueness When the Generators Form a Basis

When g_1, ..., g_m is a basis, uniqueness follows exactly as shown, and combined with the automatic compatibility described for the tensor basis rule extension, both halves of the existence-and-uniqueness pair hold unconditionally, recovering tensor multilinear basis determination as the special case of extension uniqueness applied to a basis rather than a general spanning set.

Uniqueness as the Stronger, More General Statement

Extension uniqueness, stated for an arbitrary spanning set, is strictly more general than the basis-specific uniqueness statement, since it applies even when the spanning set contains redundant generators, showing that redundancy in the specifying data affects only the existence question, addressed by compatibility, and never the uniqueness question, which holds regardless.

Practical Implications of Uniqueness

No Ambiguity Once Compatible Data Is Given

Whenever a practitioner prescribes tensor values on a spanning set and verifies compatibility, uniqueness guarantees there is exactly one tensor consistent with that data, so no further ambiguity remains about which tensor is being described, regardless of how the spanning set was chosen or how many redundant generators it contains.

Justifying Interchangeable Descriptions of the Same Tensor

Because a tensor is uniquely determined by compatible data on any spanning set, two different spanning sets, each equipped with compatible prescribed data describing the same underlying tensor, must yield identical extensions; uniqueness is what licenses treating these different descriptions as descriptions of one and the same tensor rather than as coincidentally agreeing but distinct objects.

Diagrammatic Summary

The diagram traces the uniqueness argument from agreement on the spanning generators through the vanishing of the difference tensor to the conclusion that the two tensors coincide on every possible input.