where V is a vector space over the field K, that is separately linear in each its N variables.
As the word "form" usually denotes a mapping from a vector space into its underlying field, the more general term "multilinear map" is used, when talking about a general map that is linear in all its arguments.
For N = 2, i.e. only two variables, one calls f a bilinear form.
An important type of multilinear forms are alternating multlinear forms which have the additional property of changing their sign under exchange of two arguments. This is equivalent, when K has characteristic other than 2, to saying that
,
i.e. the form vanishes if supplied the same argument twice. (The exceptional case of characteristic 2 requires more care.) Special cases of these are determinant forms and differential forms.
An alternating multilinearform on a real vector space V is a multilinearform F:V \otimes \cdots \otimes V \rightarrow \mathbb{R} such that F(x_1, \dots, x_i, x_{i+1}, \dots, x_n) = -F(x_1, \dots, x_{i+1}, x_i, \dots, x_n) for any index i.
For example, F((a_1,a_2,a_3),(b_1,b_2,b_3),(c_1,c_2,c_3))= a_1 b_2 c_3 -a_1 b_3 c_2 +a_2 b_3 c_1 - a_2 b_1 c_3 + a_3 b_1 c_2 - a_3 b_2 c_1 (3) is an alternating form on \mathbb{R}^3.
An alternating multilinearform is defined on a module in a similar...
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