= Dense subspace
{title2=$\overline D=X$}
A <dense subspace> of a <topological vector space> $X$ is a <linear subspace> $D$ with $\overline D=X$. For a <normed vector space>, every <vector> of $X$ is a <limit> of a sequence from $D$. A <continuous linear operator> into a <Hausdorff> target is uniquely determined by its values on $D$.
Existence of an extension needs additional hypotheses. A <bounded linear operator> from $D$ into a <Banach space> extends uniquely to $X$ by taking limits, as in <extension of a bounded linear operator from a dense subspace>. An incomplete target can obstruct this construction, as the <failure of dense-subspace extension into an incomplete codomain> shows.
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