Let $T: E \to F$ be a linear map between topological vector spaces $E$, $F$. If for each nonempty open set $G$, the interior of $T(G)$ is non-empty, then, $T$ is open.
Proof: $$\mathrm{Int}(T(G))= \bigcup_{U \subset T(G), U\in\tau_F} \mathcal{U}\neq \emptyset $$
Then, exist $\mathcal{U}$ such that $\mathcal{U} \subset T(G)$...
How should I proceed? Any help is appreciated. Thanks!