MathGloss

Let $V_1,\dots,V_n, W$ be vector spaces. A multilinear map is a map $$f: V_1 \times\cdots\times V_n \to W$$ such that for all $1\leq i \leq n$ and $v_j \in V_j$ with $j \neq i$, the map $$f(v_1,\dots, v_{i-1}, -, v_{i+1}, \dots, v_n):V_i \to W$$ (which varies only in the $i$th component) given by $$V_i \ni v \mapsto f(v_1,\dots, v_{i-1}, v, v_{i+1}, \dots, v_n) \in W$$ is a linear transformation.

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