nLab multilinear map

Idea

In generalization of how a bilinear map is a function of two arguments which is suitably linear in each argument, so a multilinear map is a function of any number of $n$ of arguments which is suitably linear in each of them.

For $n = 3$ one also speaks of trilinear maps, etc.

Concretely, a multilinear map of elements of abelian groups is equivlalently a single linear map out of the tensor product of all its domains. There is a multicategory Ab of abelian groups and multilinear maps between them. The bilinear maps are the binary morphisms, and the multilinear maps are the multimorphisms.

References

See also:

• Wikipedia: Multilinear map