Let $X$ and $Y$ be topological vector spaces. And $f : X \rightarrow Y$ be a linear map. For any neighborhood $W$ of $0$ in Y, there exists a neighborhood $V$ of $0$ in $X$ such that $f(V) \subset \overline W$. Then can I conclude that $f$ is continuous? The closure of $W$ makes things much trickier than I expected...
2026-03-26 16:10:41.1774541441
Continuity of a linear map between topological vector spaces
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In any Hausdorff topolgical vector space $Y$ given a neighborhood $W$ of $0$ there exists another neighborhood $W_1$ such that $\overline{W_1} \subset W$. Applying the hypothesis with $W_1$ in place of $W$ we get a neighborhood $V$ of $0$ in $X$ such that $f(V) \subset \overline {W_1}$ This gives $f(V) \subset W$.