= Dual invariance under dense inclusion
{title2=$X'\cong Z'\quad(\overline Z=X)$}
If a normed subspace $Z$ is <dense> in a <normed vector space> $X$ and carries the inherited <norm>, restriction gives an onto linear <isometry> from the <dual space> of $X$ to that of $Z$. A bounded functional on $Z$ extends uniquely by limits along sequences from $Z$; its <norm> is unchanged because its inequality passes to the limit. Thus completion does not change the <continuous> dual. For example, the subspace of sequences with $\sum j|a_j|<\infty$, equipped with the unweighted $\ell^1$ <norm>, has dual $\ell^\infty$, since it contains the <dense subspace> of finitely supported sequences. The inherited <norm> matters; the weighted <norm> would give a different dual <norm>.
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