Dual invariance under dense inclusion ($X^{\prime }\cong Z^{\prime }(\overline{Z}=X)$)

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.