Multilinear Map

In linear algebra, a multilinear map is a function of several variables that is linear separately in each variable. More precisely, a multilinear map is a function

where and are vector spaces (or modules), with the following property: for each, if all of the variables but are held constant, then is a linear function of .

A multilinear map of two variables is a bilinear map. More generally, a multilinear map of k variables is called a k-linear map. If the codomain of a multilinear map is the field of scalars, it is called a multilinear form. Multilinear maps and multilinear forms are fundamental objects of study in multilinear algebra.

If all variables belong to the same space, one can consider symmetric, antisymmetric and alternating k-linear maps. The latter coincide if the underlying ring (or field) has a characteristic different from two, else the former two coincide.