Multilinear mapping/Alternating/Definition

Alternating mapping

Let ${}K$ be a field, let ${}V$ and ${}W$ denote ${}K$-vector spaces, and let ${}n\in \mathbb {N}$. A multilinear mapping

$\Phi \colon V^{n}=\underbrace {V\times \cdots \times V} _{n{\text{-mal}}}\longrightarrow W$

is called alternating if the following holds: Whenever in ${}v=(v_{1},\ldots ,v_{n})$, two entries are identical, that is ${}v_{i}=v_{j}$ for a pair ${}i\neq j$, then

${}\Phi (v)=0\,.$