4.24 Tensor Multilinear Map Boundary

Tensor Multilinear Map Boundary is the delineation of exactly which maps are admitted as multilinear maps within the tensor-algebraic framework, separating multilinearity — linearity that holds separately in each of several arguments — from the narrower cases of linear and bilinear maps and from the broader class of maps that fail linearity in at least one slot. The boundary fixes both the domain shape, a finite Cartesian product of vector spaces and dual spaces, and the codomain, the base field or a designated target vector space, so that "multilinear map" names a precise algebraic object rather than an informal description of any function taking several vector arguments.

What Counts as Multilinear

Separate Linearity in Each Argument

A map

lies inside the boundary exactly when, for every index i and every fixed choice of the remaining arguments, the resulting function of the single remaining variable is linear: it respects vector addition and scalar multiplication in that slot alone. This condition is checked slot by slot, never across slots, which is what distinguishes multilinearity from linearity of the map as a function of the whole tuple (v_1, ..., v_k) treated as a single vector in the product space — that stronger, joint linearity is not what is required or even generally true for multilinear maps.

Arbitrary Finite Arity

The boundary places no upper limit on k, the number of arguments, beyond requiring it to be finite. A map with k = 1 is a linear map, k = 2 a bilinear map, and k = 0 is degenerate and identified with a scalar; all of these sit inside the multilinear boundary as special cases rather than as excluded neighbors. Each V_i may independently be a vector space V or its dual V*, and the pattern of how many factors are dual versus primal is what later fixes the type (p, q) of the associated tensor.

What Falls Outside the Boundary

Maps That Are Not Separately Linear

A function such as

fails the boundary condition because it is not linear in u alone, even though it takes two vector arguments; the presence of a norm, a fixed nonzero constant added outside the linear part, or any quadratic dependence on a single argument removes the map from the multilinear class regardless of how well-behaved it is otherwise.

Sesquilinear and Conjugate-Linear Maps

Over a complex vector space, a sesquilinear form is linear in one argument and conjugate-linear in the other, meaning scalar multiplication in that slot is matched with complex conjugation rather than being preserved outright. Such forms, including the standard Hermitian inner product, sit outside the multilinear boundary as defined here, since conjugate-linearity is a distinct algebraic condition from linearity; they belong to a separate sesquilinear framework built on top of, but not contained in, the multilinear one.

Maps with a Non-Field, Non-Designated Codomain

A map into an arbitrary set with no vector space structure, or into a vector space that has not been fixed in advance as part of the map's data, is excluded. The boundary requires the codomain to be either the base field F, giving an ordinary multilinear map, or a specified target vector space W, giving a vector-valued multilinear map whose own tensor identification is handled separately; an unspecified or structureless codomain removes the map from consideration entirely.

Role Within the Tensor Framework

Precondition for Tensor Identification

The multilinear boundary is what makes the identification of a map with a tensor possible in the first place: only maps satisfying separate linearity in each argument correspond to elements of a tensor product of the V_i and their duals. A map lying outside the boundary has no such tensor representative, because the universal property of the tensor product is stated in terms of multilinear maps and gives no factorization for maps that are not multilinear.

Boundary as a Filter, Not a Construction

Unlike the tensor product itself, the multilinear map boundary does not construct a new object; it classifies existing maps by testing a condition on each argument. This classifying role is what lets linear maps, bilinear maps, and multilinear forms all be treated as points inside a single boundary, differing only in arity or in whether the codomain is the field or a general target space, while maps failing separate linearity are left outside regardless of how naturally they might otherwise seem to belong to the same family.