Proposition 1

— For all $\alpha \in K$ and every point $b \in E$, the mapping $x \mapsto \alpha x + b$ of $E$ in itself is continuous. Further, if $\alpha \neq 0$, this mapping is a homeomorphism of $E$ on itself.

The second part of the proposition is a result of the fact that if $\alpha \neq 0$, then $x \mapsto \alpha^{-1}x - \alpha^{-1}b$ is the inverse mapping of $x \mapsto \alpha x + b$.