Editing Multilinear form (section)

{{Short description|Map from multiple vectors to an underlying field of scalars, linear in each argument}}

In [[abstract algebra]] and [[multilinear algebra]], a '''multilinear form''' on a [[vector space]] <math>V</math> over a [[Field (mathematics)|field]] <math>K</math> is a [[Map (mathematics)|map]]

:<math>f\colon V^k \to K</math>

that is separately <math>K</math>-[[linear]] in each of its <math>k</math> arguments.<ref>{{MathWorld|title=Multilinear Form|urlname=MultilinearForm}}</ref> More generally, one can define multilinear forms on a [[Module (mathematics)|module]] over a [[commutative ring]]. The rest of this article, however, will only consider multilinear forms on [[Dimension (vector space)|finite-dimensional]] vector spaces.

A multilinear <math>k</math>-form on <math>V</math> over <math>\R</math> is called a ('''covariant''') '''<math>\boldsymbol{k}</math>-tensor''', and the vector space of such forms is usually denoted <math>\mathcal{T}^k(V)</math> or <math>\mathcal{L}^k(V)</math>.<ref>Many authors use the opposite convention, writing <math>\mathcal{T}^k(V)</math> to denote the contravariant ''k''-tensors on <math>V</math> and <math>\mathcal{T}_k(V)</math> to denote the covariant ''k''-tensors on <math>V</math>.</ref>