Let $X$ be a topological vector space and let $f$ be a linear functional on $X$. If $f$ is continuous, how to show $f$ is bounded in some neighborhood of $0$?
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2026-03-25 12:50:49.1774443049
Let $X$ be a TVS and let $f$ be a linear functional on $X$. If $f$ is continuous, how to show $f$ is bounded in some neighborhood of $0$?
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We can assume $f \neq 0$. Or else any neighbourhood of $0$ will do.
$O = f^{-1}[(-1,1)]$ is an open neighbourhood of $0$ by continuity of $f$ (and $f(0) = 0$ of course). And clearly $f$ is bounded on $O$.