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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!

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    Where did you use the vector space structure?2012-12-03

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