14.1.5 Tensor Map Product Verification Scope

Tensor Map Product Verification Scope ensures correctness in tensor algebra by validating operations within structured mathematical frameworks.

Tensor Map Product Verification Scope is the collection of checks that must be carried out to confirm that a proposed assignment on elementary tensors genuinely defines a linear map on a tensor product of maps, together with the range of properties that such a verification is expected to certify.

Purpose of the Verification Scope

What Must Be Verified

A formula given only on elementary tensors is not automatically a well-defined map on the tensor product, since elementary tensors satisfy relations coming from bilinearity, and a candidate rule must respect every one of those relations. The verification scope covers exactly the checks needed to confirm that a rule of the form

v \otimes w \mapsto f (v) \otimes g (w)

extends to a genuine linear map on the whole tensor product.

Bilinearity Checks

Verification requires confirming additivity in each argument separately,

(v_{1} + v_{2}) \otimes w \mapsto f (v_{1}) \otimes g (w) + f (v_{2}) \otimes g (w),

together with the analogous identity in the second argument, and homogeneity in each argument,

(λ v) \otimes w \mapsto λ (f (v) \otimes g (w)) .

Only once every relation of this kind is confirmed can the assignment be regarded as descending to a single well-defined map on the tensor product rather than merely a formula on generators.

Verification via the Universal Property

Reducing Verification to a Bilinear Map

Rather than checking every relation among elementary tensors directly, the verification scope is usually discharged by exhibiting the underlying assignment as a bilinear map on the product of the two domains and invoking the universal property of the tensor product. Once bilinearity of

β (v, w) = f (v) \otimes g (w)

is verified, the existence of a unique linear map on the tensor product agreeing with this rule on elementary tensors is guaranteed by the universal property itself, and no further relation-by-relation check is required.

Uniqueness Verification

Verification scope also includes confirming that the induced linear map is unique, which follows from the fact that elementary tensors span the tensor product: any two linear maps agreeing on all elementary tensors must agree everywhere, so uniqueness is verified as soon as spanning is established rather than requiring a separate argument.

Verification of Derived Properties

Verifying Compatibility with Composition

Beyond existence and uniqueness, the verification scope extends to confirming the identities that make the tensor product of maps well behaved under composition,

(f^{'} \otimes g^{'}) \circ (f \otimes g) = (f^{'} \circ f) \otimes (g^{'} \circ g) .

Such an identity is verified by checking that both sides agree on elementary tensors, and then invoking the spanning property of elementary tensors to conclude agreement everywhere, exactly as in the existence argument.

Verifying Rank and Injectivity Claims

The verification scope also covers computational claims such as the rank formula

rank (f \otimes g) = rank (f) \cdot rank (g),

which is verified by reducing to the Kronecker product description in bases adapted to the images of f and g, and confirming that the resulting block matrix has the claimed rank through elementary row and column operations.

Boundaries of the Verification Scope

What Falls Outside

The verification scope does not extend to checking properties of f and g individually, such as their own linearity, which must be established beforehand as a precondition rather than as part of verifying the tensor product construction. It also does not extend to verifying properties of the ambient tensor product itself, such as its existence or its universal property, which are assumed as already established facts about the tensor product construction on which the verification of the map is built.