Let \(m,n_1,n_2,\ldots,n_k\) be nonnegative integers. A function \(L\colon \R^{n_1}\times \R^{n_2}\times \cdots \times
\R^{n_k}\to \R^m \) is called multilinear if
\(L((\mathbf{x}_1,\mathbf{x}_2,\ldots,\mathbf{x}_k)+
(\mathbf{y}_1,\mathbf{y}_2,\ldots,\mathbf{y}_k))
= L(\mathbf{x}_1,\mathbf{x}_2,\ldots,\mathbf{x}_k) +
L(\mathbf{y}_1,\mathbf{y}_2,\ldots,\mathbf{y}_k)\text{,}\) and
for all \(k\)-tuples of vectors \((\mathbf{x}_1,\mathbf{x}_2,\ldots,\mathbf{x}_k),(\mathbf{y}_1,\mathbf{y}_2,\ldots,\mathbf{y}_k)\) in \(\R^n\) and \(k\)-tuples of real numbers \(\alpha_1,\alpha_2,\ldots,\alpha_k\text{.}\)
