14.5 Tensor Linear Functional Product Structure

Tensor Linear Functional Product Structure is the organization of the data specific to the functional case of the tensor product of maps, in which both maps of the factor map pair are linear functionals, producing an induced functional on the tensor product together with the identifications this specialization supports between tensor products of functionals and the dual of the tensor product.

Specializing the General Structure to Functionals

The Functional Factor Map Pair

Where the general structure allows a factor map pair with two arbitrary target spaces, the functional product structure restricts to

with F the common field of scalars, so the factor space relation identifies both codomain factors with the same one-dimensional space F, collapsing the codomain tensor product F tensor F to F itself through scalar multiplication.

The Reduced Codomain

Because the codomain tensor product F tensor F is canonically identified with F, the functional product structure differs from the general structure in that its output is a genuine scalar rather than an elementary tensor of two separate target spaces, giving

Assembly of the Structure

From Pair and Relation to a Bilinear Form

The functional product structure assembles the factor map pair into the bilinear form

whose factorization through the universal property of the tensor product produces the induced functional phi tensor psi on V tensor W.

Identification with the Dual of the Tensor Product

The functional product structure supports the canonical map

sending phi tensor psi to the functional described above, an isomorphism in the finite-dimensional case, giving the functional product structure its role as a bridge between elementary tensors of functionals and the abstract dual space of the tensor product.

Internal Structure

Bilinearity in the Factor Map Pair

The assignment sending a pair of functionals to their tensor product functional is itself bilinear: linear in phi for fixed psi, and linear in psi for fixed phi, since

matching the general bilinearity of the operation sending a factor map pair to its tensor product, specialized here to functionals valued in the same field F.

Dual Basis Structure

Given bases of V and W with dual bases, the functional product structure assigns to each pair of dual basis vectors a functional on V tensor W, and these functionals, ranging over all such pairs, form the dual basis of the induced basis of elementary tensors on V tensor W, giving the functional product structure a direct combinatorial description in coordinates.

Behavior Under Composition and Pullback

Pullback Along Operators

If f is an operator on V and g is an operator on W, the functional product structure interacts with pullback through

reducing the pullback of a tensor product of functionals to a tensor product of the individually pulled back functionals, matching the general composition identity of tensor products of maps specialized to this case.

Connection to Rank-One Bilinear Forms

Every elementary tensor phi tensor psi corresponds, under the functional product structure, to a bilinear form of rank at most one on V times W, and general elements of the tensor product of duals correspond to bilinear forms expressible as finite sums of such rank-one forms, linking the functional product structure directly to the classification of bilinear forms by rank.