14.21.2 Tensor Map Product Functional Construction Role

Tensor Map Product Functional Construction Role explains how tensor maps compose functionally, defining their role in algebraic operations.

Tensor Map Product Functional Construction Role is the use of the tensor product of maps, specialized to the case where each factor is a linear functional, to build new linear functionals on a tensor product space out of linear functionals on the individual factor spaces, giving a systematic way of producing bilinear forms and multilinear functionals from simpler, single-variable data.

The Specialization to Functionals

Functionals as Maps Into the Base Field

A linear functional on $V$ is a linear map $φ : V \to K$ into the base field, viewed as a one-dimensional vector space over itself. Taking $V^{'} = K$ and $W^{'} = K$ in the general construction of $f \otimes g$ specializes it to the tensor product of two functionals $φ : V \to K$ and $ψ : W \to K$ , producing

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

Identifying the Target With the Base Field

Since $K \otimes K$ is canonically identified with $K$ itself, via $α \otimes β \mapsto α β$ , the tensor product of two functionals is naturally regarded as a single linear functional

φ \otimes ψ : V \otimes W \to K, (φ \otimes ψ) (v \otimes w) = φ (v) \cdot ψ (w)

recovering the familiar formula for a decomposable functional on a tensor product space as a product of two scalars.

Constructing Bilinear Forms

Every Decomposable Functional Is a Bilinear Form

Composing $φ \otimes ψ$ with the canonical bilinear map $V \times W \to V \otimes W$ sending $(v, w)$ to $v \otimes w$ produces the bilinear form

B (v, w) = φ (v) ψ (w)

on $V \times W$ directly, without any further construction; this is the primary use of the functional construction role, converting a pair of single-variable functionals into a genuinely two-variable bilinear form.

Sums Produce General Bilinear Forms

A single decomposable form $φ \otimes ψ$ is rarely the most general bilinear form available, but a finite sum

B = \sum_{k = 1}^{r} φ_{k} \otimes ψ_{k}

of such decomposable functionals reaches every bilinear form on finite-dimensional $V$ and $W$ , which is precisely the statement that the natural map $V^{*} \otimes W^{*} \to {(V \otimes W)}^{*}$ built from tensor products of functionals is surjective, and in fact an isomorphism, in finite dimensions.

Role in Building the Dual Space Isomorphism

The Canonical Map on Dual Spaces

The assignment $(φ, ψ) \mapsto φ \otimes ψ$ , extended by linearity, defines the canonical linear map

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

which is exactly the tensor map product construction applied to the special case of functionals; the fact that this map is a bijection when $V$ and $W$ are finite-dimensional is the reason the dual of a tensor product is identified with the tensor product of the duals.

Injectivity From Rank Considerations

Injectivity of this map, at least on nonzero simple tensors $φ \otimes ψ$ , follows from the rank formula $rank (φ \otimes ψ) = rank (φ) \cdot rank (ψ)$ established for the tensor product of maps generally, since a nonzero functional has rank one, giving a nonzero decomposable functional whenever both $φ$ and $ψ$ are nonzero.

Constructing Higher Multilinear Functionals

Iterating the Construction

Tensoring three or more functionals, $φ_{1} \otimes φ_{2} \otimes φ_{3}$ , applies the tensor map product construction repeatedly, each time in the functional-valued special case, to build multilinear functionals of three or more variables from single-variable functionals, extending the bilinear form construction to forms of arbitrary arity.

Practical Role in Constructing Test Functionals

This construction is the standard way to produce concrete test functionals on a tensor product space when verifying properties of specific tensors, since evaluating $(φ_{1} \otimes \dots \otimes φ_{k})$ on a candidate tensor reduces immediately to evaluating each $φ_{i}$ on a single vector and multiplying the results, a far simpler computation than evaluating a general element of the dual space directly.