Proposition 1

Every continuous homomorphism $f$ of the additive group $\mathbf{R}^m$ into the additive group $\mathbf{R}^n$ is a linear mapping of $\mathbf{R}^m$ into $\mathbf{R}^n$.

It is enough to show that $f(tx) = tf(x)$ for all $x \in \mathbf{R}^n$ and all $t \in \mathbf{R}$. The argument is the same as that of Proposition 5 of Chapter V, § i, no. 3 if we replace $x$ by $x$ and $\mathbf{R}$ by $\mathbf{R}^m$.