14.5.1 Tensor Linear Functional Product Factor Selection

Tensor Linear Functional Product Factor Selection is the process of choosing which linear functional acts on which tensor factor when forming a functional product phi tensor psi, together with the consequences of selecting a functional for only one factor while leaving the other factor evaluated by a fixed reference vector or by a designated coevaluation.

The Basic Selection

Choosing a Functional per Factor

Given functionals phi on V and psi on W, factor selection assigns phi to the first tensor factor and psi to the second, producing

Reversing the selection produces psi tensor phi, a functional on the differently ordered tensor product W tensor V, which agrees with phi tensor psi only after composing with the canonical swap isomorphism between V tensor W and W tensor V.

Selecting a Functional for Only One Factor

A factor selection may assign a functional to only one factor while leaving the other unassigned, in which case the unassigned factor is typically paired with the trivial one-dimensional identity structure of the field, effectively evaluating that factor through a fixed choice rather than through a general functional.

Consequences of Partial Selection

Fixing One Vector Instead of Selecting a Functional

Rather than selecting a functional for the second factor, one may instead fix a specific vector w in W, producing the partial map

a map from V into F tensor W, rather than a functional into F, illustrating that fixing a vector instead of selecting a second functional changes the nature of the resulting map from a scalar-valued functional to a vector-valued map.

Selection Restricted to a Subspace

If the factor selection for phi is made only on a subspace U of V rather than on all of V, the resulting functional phi tensor psi is correspondingly defined only on U tensor W, and extending this restricted selection back to a functional on the whole of V tensor W requires first extending phi itself to a functional on all of V, since the factor selection cannot supply values on the complementary elementary tensors without such an extension.

Selection and Bilinear Form Correspondence

Selection Determining Rank-One Contributions

Selecting phi and psi for a single functional product corresponds, under the identification of functional products with bilinear forms, to selecting a single rank-one bilinear form on V times W; combining several such selections by summing multiple functional products corresponds to building a bilinear form of higher rank as a sum of rank-one contributions, each contribution determined by its own factor selection.

Selection and the Gram Matrix

Once bases are fixed, selecting phi with coordinate row vector a and psi with coordinate row vector b corresponds to selecting the rank-one matrix formed as the outer product of a and b, so factor selection in this structure corresponds directly to choosing the two vectors whose outer product forms the Gram matrix of the resulting functional.

Selection Under Dualization and Pullback

Selection Compatible with Pullback

If f is an operator on V and g is an operator on W, the factor selection for the pullback functional

matches the selection of phi composed with f for the first factor and psi composed with g for the second factor, so factor selection is preserved consistently when the functional product is pulled back along an operator product.

Selection of Dual Basis Functionals

A distinguished factor selection uses dual basis functionals for both phi and psi, producing the induced dual basis of V tensor W directly; selecting any other pair of functionals instead produces a functional expressible as a linear combination of these dual basis selections, so the dual basis selection serves as the canonical reference against which all other factor selections in this structure are measured.