Multilinear form

In abstract algebra and multilinear algebra, a multilinear form on a vector space ${\displaystyle V}$ over a field ${\displaystyle K}$ is a map

${\displaystyle f\colon V^{k}\to K}$

that is separately ${\displaystyle K}$-linear in each of its ${\displaystyle k}$ arguments.^[1] More generally, one can define multilinear forms on a module over a commutative ring. The rest of this article, however, will only consider multilinear forms on finite-dimensional vector spaces.

A multilinear ${\displaystyle k}$-form on ${\displaystyle V}$ over ${\displaystyle \mathbb {R} }$ is called a (covariant) ${\displaystyle {\boldsymbol {k}}}$-tensor, and the vector space of such forms is usually denoted ${\displaystyle {\mathcal {T}}^{k}(V)}$ or ${\displaystyle {\mathcal {L}}^{k}(V)}$.^[2]