Proposition 3

Every continuous homomorphism of $\mathbf{R}^m$ into $\mathbf{T}^n$ is of the form $x \to \varphi(u(x))$, where $\varphi$ is the canonical homomorphism of $\mathbf{R}^n$ onto $\mathbf{T}^n$ (identified with $\mathbf{R}^n/\mathbf{Z}^n$) and $u$ is a linear mapping of $\mathbf{R}^m$ into $\mathbf{R}^n$.

Let $f$ be a continuous homomorphism of $\mathbf{R}^m$ into $\mathbf{T}^n$. We shall show that there is a linear mapping $u$ of $\mathbf{R}^m$ into $\mathbf{R}^n$ such that the homomorphisms $x \to f(x)$ and $x \to \varphi(u(x))$ coincide at all points of a neighbourhood of $o$ in $\mathbf{R}^m$; the proposition will then follow from Corollary 1 to Proposition 2 of no. 2. Now let $V$ be a neighbourhood of $o$ in $\mathbf{R}^n$ such that $\varphi$, restricted to $V$, is a local isomorphism of $\mathbf{R}^n$ with $\mathbf{T}^n$, and let $\psi$ be the inverse of $\varphi$, defined on $\varphi(V)$. Since $f$ is continuous, $V' = \overline{f}^{-1}(\varphi(V))$ is a neighbourhood of $o$ in $\mathbf{R}^m$; the mapping

restricted to $V'$, is a continuous mapping of $V'$ into $\mathbf{R}^n$ such that $\psi(f(x+y)) = \psi(f(x)) + \psi(f(y))$ for each pair of points $x, y$ in $\mathbf{R}^m$ such that $x \in V', y \in V'$ and $x + y \in V'$; hence (no. 2, Corollary 1 to Proposition 2) this mapping coincides with a well-determined continuous homomorphism $u$ of $\mathbf{R}^m$ into $\mathbf{R}^n$ at all points of a neighbourhood $W$ of o in $\mathbf{R}^m$. By Proposition 1 (no. 1) $u$ is a linear mapping of $\mathbf{R}^m$ into $\mathbf{R}^n$; for all $x \in W$ we have therefore $f(x) = \varphi(u(x))$, which completes the proof.