4.24.3 Tensor Multilinear Form Boundary

The Tensor Multilinear Form Boundary marks where linear operations in tensor algebra transition to higher-order interactions.

Tensor Multilinear Form Boundary is the delineation of which multilinear maps are classified as multilinear forms, restricting the general multilinear map boundary to the single case where the codomain is fixed to be the base field F itself rather than an arbitrary designated target vector space W. Every multilinear form is a multilinear map, but the converse holds only within this narrower boundary; a vector-valued multilinear map with W ≠ F lies inside the multilinear map boundary while lying outside the multilinear form boundary.

The Defining Restriction

Codomain Fixed to the Base Field

A multilinear map

ω : V_{1} \times \dots \times V_{k} \to F

lies inside the multilinear form boundary precisely when its codomain is the base field, with no further condition imposed beyond the multilinearity already required by the general multilinear map boundary. The restriction is entirely on the codomain; the domain factors V_i remain free to be vector spaces or their duals in any combination, exactly as in the unrestricted multilinear map boundary.

Identification with Elements of a Dual Tensor Power

Because the codomain is the field, a multilinear form on V_1 × ⋯ × V_k corresponds, via the tensor product factorization boundary, to an element of

V_{1}^{*} \otimes \dots \otimes V_{k}^{*}

when each V_i is finite-dimensional, since a linear functional on V_i is exactly an element of V_i*. This places multilinear forms squarely among the objects the tensor framework calls tensors of purely covariant type, in contrast to vector-valued multilinear maps, whose tensor identification requires the extra W factor described separately.

What Falls Outside the Boundary

Vector-Valued Multilinear Maps

A map such as the cross product

\times : ℝ^{3} \times ℝ^{3} \to ℝ^{3}

is bilinear, and therefore multilinear, but is not a multilinear form, because its codomain is ℝ³ rather than the base field ℝ; it is excluded from this boundary even though it satisfies every other condition of multilinearity.

Maps to the Field That Are Not Multilinear

A map into the base field that fails separate linearity in some argument, such as a quadratic form, is excluded on the multilinearity condition inherited from the general multilinear map boundary, independent of the codomain restriction; the form boundary narrows the codomain but does not relax the multilinearity requirement.

Relation to Adjacent Boundaries

Special Case of Tensor Multilinear Map Boundary

Tensor Multilinear Form Boundary is a strict subset of Tensor Multilinear Map Boundary, obtained by setting the target space equal to the field in the general definition; every requirement of the map boundary applies unchanged, with the form boundary adding only the single codomain restriction.

Bilinear Forms and Linear Functionals as Low-Arity Instances

At k = 1, a multilinear form is exactly a linear functional, an element of V*; at k = 2, it is exactly a bilinear form, matching the bilinear map boundary restricted to field-valued codomain. The multilinear form boundary generalizes both without altering their content, treating them as the k = 1 and k = 2 cases of a single arity-indexed family.