1.2.20 Multilinear Map Definition

Multilinear Map Definition is the characterization of a multilinear map as a function of several vector arguments, drawn from one or more vector spaces defined over a common scalar field, that is linear in each argument separately whenever every other argument is held fixed. It generalizes the bilinear map from exactly two arguments to any finite number of arguments, and it is the general notion whose associated tensor product construction leads directly to the formal definition of a tensor of arbitrary rank.

The General Definition

A function taking a fixed number of vector arguments, each drawn from a vector space over a common scalar field, and producing a result in some vector space, is called multilinear if, for each of its arguments in turn, fixing every other argument and varying only that one produces an ordinary linear map. This condition must hold independently and simultaneously for every argument position: linearity in one argument alone, without regard to the others, does not suffice for the function to be considered multilinear.

The expression above states linearity in the i-th argument of an n-argument multilinear map, with every other argument held fixed; the full definition of multilinearity requires this condition to hold for each argument position from the first to the n-th.

Naming Conventions by Number of Arguments

Multilinear maps taking a specific small number of arguments are given their own names: a multilinear map of one argument is simply a linear map, a multilinear map of two arguments is a bilinear map, and a multilinear map of three arguments is a trilinear map. Beyond three arguments, the maps are generally referred to collectively as multilinear maps of the corresponding number of arguments, or as k-linear maps, where k denotes the number of arguments involved.

Multilinear Forms

A multilinear map whose output lies in the underlying scalar field, rather than in some other vector space, is called a multilinear form. Multilinear forms are further classified according to their behavior under permutation of their arguments: a symmetric multilinear form is unchanged by any reordering of its arguments, while an alternating multilinear form changes sign whenever two of its arguments are swapped. Symmetric multilinear forms give rise to the symmetric algebra, and alternating multilinear forms give rise to the exterior algebra, both constructed as quotients of the full tensor algebra by relations reflecting these respective symmetries.

Multilinear Maps Are Not Linear on the Combined Space

Just as with bilinear maps, a multilinear map of several arguments is not, in general, linear when its arguments are combined into a single vector belonging to the direct sum of the spaces involved, because multilinearity allows for multiplicative interaction between the different arguments that an ordinary linear map on the combined space could not capture. This is precisely why the tensor product construction becomes necessary: it produces a new vector space on which the equivalent information carried by a multilinear map can instead be represented by an ordinary linear map.

From Multilinear Maps to Tensors

The universal property of the tensor product extends the bilinear case to the fully general multilinear case: for any multilinear map defined on a collection of vector spaces, there exists a unique linear map from the tensor product of those spaces that reproduces the original multilinear map's values. This correspondence is what allows a tensor of type (p, q) to be defined equivalently either as an element of the appropriate tensor product space, or as a multilinear map taking p covectors and q vectors as arguments and producing a scalar, since the two formulations are connected precisely through this universal property applied to multilinear maps of the corresponding number of arguments.