14.2.2 Tensor Linear Functional Product Area

Explore how tensor linear functionals interact through product areas in algebraic structures and their applications in mathematical frameworks.

Tensor Linear Functional Product Area is the branch of study concerned with tensor products of linear maps in the special case where the maps involved are linear functionals, meaning linear maps from a vector space to its field of scalars, and with the identifications this specialization produces between tensor products of maps and tensor products of dual spaces.

Distinguishing the Functional Case

Functionals as the Common Setting

Where the general tensor product of maps allows f and g to take values in arbitrary target spaces, the functional product area restricts attention to

φ : V \to F

ψ : W \to F

where F denotes the field of scalars, so that the induced map

φ \otimes ψ : V \otimes W \to F \otimes F

lands in F tensor F, a space canonically identified with F itself through multiplication of scalars, so that phi tensor psi is naturally regarded as a linear functional on V tensor W.

Canonical Identification with the Dual of the Tensor Product

This area is built around the canonical linear map

V^{*} \otimes W^{*} \to {(V \otimes W)}^{*},

sending phi tensor psi to the functional acting on elementary tensors by

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

In the finite-dimensional case this canonical map is an isomorphism, so every functional on the tensor product arises from such tensor products of functionals, though not every such functional is itself an elementary tensor of two functionals.

Behavior on Bases and Coordinates

Dual Basis Pairing

If a basis of V and a basis of W are chosen together with their dual bases, then the functionals phi tensor psi built from dual basis vectors form a dual basis of V tensor W with respect to the corresponding basis of elementary tensors, so that

(e_{i}^{*} \otimes e_{j}^{*}) (e_{k} \otimes e_{l}) = δ_{ik} δ_{jl},

with the right-hand side equal to one when the indices match on both factors and zero otherwise. This identity is the coordinate expression of the canonical isomorphism between the tensor product of duals and the dual of the tensor product.

Coordinates as Products of Coordinates

Consequently, the coordinates of an elementary tensor v tensor w with respect to the induced basis are exactly the products of the coordinates of v with respect to the basis of V and the coordinates of w with respect to the basis of W, matching the block structure of the Kronecker product when phi and psi are represented as row vectors.

Behavior Under Composition and Pullback

Precomposition by Operators

If f is an operator on V and g is an operator on W, the functional obtained by tensoring phi and psi and then precomposing with f tensor g satisfies

(φ \otimes ψ) \circ (f \otimes g) = (φ \circ f) \otimes (ψ \circ g),

expressing the pullback of a tensor product of functionals along a tensor product of operators as the tensor product of the pullbacks along the individual operators, a direct specialization of the general composition rule for tensor products of maps.

Relation to Bilinear Forms

Every elementary tensor phi tensor psi corresponds to the bilinear form on V times W sending the pair (v, w) to phi(v) psi(w), and sums of such elementary tensors correspond to the general bilinear forms obtainable as finite sums of products of one functional on each factor, connecting the functional product area directly to the classification of bilinear forms of finite rank.