14.5.4 Tensor Linear Functional Product Evaluation Rule

The Tensor Linear Functional Product Evaluation Rule defines how tensors act on linear functionals via index contraction and pairing.

Tensor Linear Functional Product Evaluation Rule is the rule specifying how the induced functional phi tensor psi is evaluated on elements of the tensor product V tensor W, reducing evaluation on elementary tensors to a product of scalars and extending that reduction by linearity to arbitrary elements.

Evaluation on Elementary Tensors

The Elementary Evaluation Rule

For functionals phi on V and psi on W, the evaluation rule on an elementary tensor v tensor w is

(φ \otimes ψ) (v \otimes w) = φ (v) \cdot ψ (w),

fixing the evaluation as an ordinary product of the two scalar values obtained by evaluating phi at v and psi at w separately, with no interaction between the two evaluations beyond this final multiplication.

Order of Evaluation Does Not Matter

Because the two evaluations phi(v) and psi(w) each produce ordinary scalars in a commutative field, the order in which they are computed does not affect the evaluation rule, so phi(v) times psi(w) equals psi(w) times phi(v), and the evaluation rule may be computed by evaluating either functional first.

Evaluation on General Elements

Extension to Sums of Elementary Tensors

For a general element expressed as a finite sum of elementary tensors, the evaluation rule extends by linearity to

(φ \otimes ψ) (\sum_{i} v_{i} \otimes w_{i}) = \sum_{i} φ (v_{i}) \cdot ψ (w_{i}),

a sum of ordinary scalar products, one for each elementary tensor appearing in the given decomposition of the element.

Independence from the Decomposition

Since a general element of V tensor W typically admits several distinct decompositions into sums of elementary tensors, the evaluation rule is meaningful only because the resulting scalar sum is guaranteed to be the same regardless of which decomposition is used, a guarantee provided by the universal property underlying the construction of phi tensor psi rather than by any direct check performed decomposition by decomposition.

Evaluation Rule in Coordinates

Coordinate Reduction

If v has coordinates given by a column vector c with respect to a basis of V, and w has coordinates given by a column vector d with respect to a basis of W, and phi and psi have coordinate row vectors a and b with respect to the corresponding dual bases, the evaluation rule reduces to

(φ \otimes ψ) (v \otimes w) = (a c) \cdot (b d),

expressing the evaluation rule as the product of two ordinary matrix products, each producing a single scalar before the final multiplication.

Evaluation via the Kronecker Product Row Vector

Equivalently, the evaluation rule can be computed in a single step as the product of the Kronecker product row vector a tensor b with the Kronecker product column vector c tensor d, giving

(φ \otimes ψ) (v \otimes w) = (a \otimes b) (c \otimes d),

an ordinary matrix product of a single row against a single column, matching the general Kronecker product description of a tensor product of maps specialized to the case of two functionals.

Consequences of the Evaluation Rule

Vanishing Evaluation

The evaluation rule gives zero whenever either phi(v) or psi(w) is zero, so the kernel of phi tensor psi contains every elementary tensor built from a vector in the kernel of phi paired with any vector in W, or from any vector in V paired with a vector in the kernel of psi, directly identifying a large family of elements on which the evaluation vanishes without further computation.

Consistency with Bilinear Form Evaluation

The evaluation rule for phi tensor psi on an elementary tensor v tensor w matches exactly the evaluation of the corresponding bilinear form at the pair (v, w), confirming that the identification of functional products with bilinear forms is compatible with how each side is actually evaluated, not merely how each side is defined.