Multilinear mapping/K/Definition

Multilinear mapping

Let ${}K$ be a field, and let ${}V_{1},\ldots ,V_{n}$ and ${}W$ be vector spaces over ${}K$. A mapping

$\Phi \colon V_{1}\times \cdots \times V_{n}\longrightarrow W$

is called multilinear if, for every ${}i\in \{1,\ldots ,n\}$ and every ${}(n-1)$-tuple ${}(v_{1},\ldots ,v_{i-1},v_{i+1},\ldots ,v_{n})$ with ${}v_{j}\in V_{j}$, the induced mapping

$V_{i}\longrightarrow W,v_{i}\longmapsto \Phi (v_{1},\ldots ,v_{i-1},v_{i},v_{i+1},\ldots ,v_{n}),$

is ${}K$-linear.