Multilinear map

Definition

Let $F$ be a field, let $n \ge 1$, and let $V_1, \dots, V_n$ and $W$ be vector spaces over $F$. A map $f \colon V_1 \times \cdots \times V_n \to W$ is multilinear when it is linear in each variable separately. That is, for each $i$ and each fixed choice of $v_j \in V_j$ for $j \ne i$, the map $v \mapsto f(v_1, \dots, v_{i-1}, v, v_{i+1}, \dots, v_n)$ is a linear map $V_i \to W$.