Corollary

— If $A$ is an open (resp. closed) set in $E$, then $\alpha A$ is open (resp. closed) in $E$ for every $\alpha \neq 0$ in $K$.

Let $E$ and $F$ be two topological vector spaces on the same topological division ring $K$. A bijection $f$ of $E$ on $F$ is an isomorphism of the topological vector space $E$ on the topological vector space $F$ if and only if $f$ is linear and bicontinuous. In particular, if $\gamma \neq 0$ belongs to the centre of $K$, the homothety $x \mapsto \gamma x$ is an automorphism of the topological vector space structure of $E$.