Let $f : \prod_{n = 1}^N V_n \to W$ be a
D1423: Multilinear map such that
| (i) |
$0_1, \ldots, 0_N, 0_W$ are each the D737: Zero vector in $V_1, \ldots, V_N, W$, respectively
|
Let $x = (x_1, \ldots, x_N) \in \prod_{n = 1}^N V_n$ be a
D1129: Vector in $\prod_{n = 1}^N V_n$ such that
| (i) |
\begin{equation}
\exists \, n \in 1, \ldots, N :
x_n = 0_n
\end{equation}
|
Let $R$ be the division ring over which $\prod_{n = 1}^N V_n$ and $W$ are vector spaces and let $0_R$ be the additive identity in $R$. Since $0_n = 0_R 0_n$, since $0_R w = 0_W$ for every $w \in W$, and since $f$ is homogeneous in each argument, then
\begin{equation}
\begin{split}
f(x)
= f(x_1, \ldots, x_N)
& = f(x_1, \ldots, x_n, \ldots, x_N) \\
& = f(x_1, \ldots, 0_n, \ldots, x_N) \\
& = f(x_1, \ldots, 0_R 0_n, \ldots, x_N)
= 0_R f(x_1, \ldots, 0_n, \ldots, x_N)
= 0_W
\end{split}
\end{equation}
$\square$
