Then $f$ is a linear map from $V$ to $W$ over $R$ if and only if
\begin{equation}
\forall \, N \in \{ 1, 2, 3, \ldots \} :
\forall \, x \in V^N :
\forall \, r \in R^N :
f \left( \sum_{n = 1}^N r_n x_n \right) = \sum_{n = 1}^N r_n f(x_n)
\end{equation}
