If $X$ is a normed linear space and $X^*$ be its completion, consider a linear functional $f$ belonging to $X'$ which is a closed map. By Hahn-Banach extension there exists a linear functional $f_1 \colon X^*\to\mathbb R$, s.t. $f_1$ restricted to $X$ is $f$ and $||f_1||=||f||$. Question: is $f_1$ also closed?
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