14.20 Tensor Map Product Verification Procedure

Tensor Map Product Verification Procedure is the systematic sequence of checks used to confirm that a given linear map between two tensor product spaces is genuinely equal to the tensor product map $f\otimes g$ of two specified linear maps, rather than merely resembling it, by reducing the verification to a finite check on simple tensors or on a single basis.

Why Verification Is Needed

The Gap Between a Formula and a Proof

A candidate map $h:V\otimes W\to V^{\prime }\otimes W^{\prime }$, produced for instance from a component formula or from an ad hoc construction, is not automatically the tensor product of $f$ and $g$ just because it looks similar; equality of linear maps on the entire space $V\otimes W$ must be established, and the verification procedure exists because checking this directly, element by element, is neither necessary nor practical.

The Reduction Principle Underlying the Procedure

The procedure exploits the fact that $V\otimes W$ is spanned by simple tensors, and that a linear map is entirely determined by its values on any spanning set; consequently, verifying agreement on a spanning set, rather than on every element, suffices to establish equality of two linear maps everywhere.

The Verification Procedure

Step One: Confirm Linearity of the Candidate

Before any comparison is meaningful, confirm that $h$ is itself linear, since the tensor product map is defined only as a linear map and any comparison to a non-linear candidate is not applicable.

Step Two: Check Agreement on Simple Tensors

Verify that

holds for every simple tensor $v\otimes w$, with $v$ ranging over all of $V$ and $w$ over all of $W$, not merely over a chosen basis at this stage.

Step Three: Invoke the Spanning Property

Since simple tensors span $V\otimes W$, and two linear maps that agree on a spanning set agree everywhere, agreement on all simple tensors, established in step two, already implies $h=f\otimes g$ as maps on the whole space; no further check on general linear combinations is logically required, though it may still be performed as a consistency check.

A Sharper Reduction Using a Basis

Reducing Further to Basis Input Elements

Because a basis for $V\otimes W$ can be built from basis input elements $e_i\otimes f_j$, and a basis is in particular a spanning set, step two can be replaced by the finitely many checks

for the $nm$ pairs $(i,j)$, which is a strictly finite verification whenever $V$ and $W$ are finite-dimensional, in contrast with the uncountably many simple tensors checked in step two.

Why the Finite Check Is Equally Valid

This reduction is valid because linearity of both $h$ and of the map $(v,w)\mapsto f\left(v\right)\otimes g\left(w\right)$ extends agreement on the finite basis input elements to agreement on all simple tensors by bilinearity, and thence to all of $V\otimes W$ by the spanning property, so the finite check and the check over all simple tensors are logically equivalent.

Common Pitfalls the Procedure Guards Against

Checking Only a Proper Subset of Basis Input Elements

Verifying the formula on some, but not all, of the $nm$ basis input elements does not establish equality, since the unchecked basis input elements could carry arbitrary discrepancies; the procedure requires the check to run over the complete set of pairs $(i,j)$.

Confusing Agreement on Simple Tensors With Agreement of Formulas

A candidate map might agree with $f\otimes g$ on every simple tensor while being described by a superficially different-looking formula; the procedure only certifies equality of the maps as functions, not similarity of the expressions used to define them, so two verified-equal maps may still admit very different-looking, but mathematically equivalent, formulas.

Neglecting to Check Linearity of the Candidate First

Skipping step one and proceeding directly to compare values on simple tensors can produce a false sense of verification if $h$ is not linear, since agreement on a spanning set only forces equality of linear maps; without confirmed linearity, agreement on simple tensors alone does not extend to the rest of $V\otimes W$.