Let $\overline{X}$ be Banach space and $X$ be dense subset of it. Show that dual space of X and dual space of $\overline{X}$ are isomorphic.
Why these are isomorphic?
I don't know how to prove it.
2026-03-29 20:17:41.1774815461
The dual space of normed vector space $X$ is isomorphic to the dual of its completion
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Apparently, $X$ is a dense subspace of $\bar X$, in order to define its dual.
Clearly, if $\ell\in X^*$, then $\ell$ extends uniquely, by a standard density argument to an $\bar\ell\in \bar X^*$, and clearly $\|\bar\ell\|=\|\ell\|$. Inversely, if $\bar\ell\in\bar X^*$, then its restriction to $X$ is a bounded linear functional on $X$.