Proposition 5

Let $E, F, G$ be three finite-dimensional vector spaces over the field $\mathbf{R}$. Then every bilinear mapping () $f$ of $E \times F$ into $G$ is continuous.*

We may suppose that $E = \mathbf{R}^m, F = \mathbf{R}^n, G = \mathbf{R}^p$; it is enough to show that the coordinates in $\mathbf{R}^p$ of $f(x, y)$ are continuous functions of $(x, y) \in E \times F$ (Chapter I, § 4, no. 1, Proposition 1). In other words, it is enough to show that every bilinear form $g$ is continuous on $E \times F$; and this is immediate, since $g(x, y)$ is a polynomial in the coordinates of $x$ and $y$.