Dense subspace ($\overline{D}=X$)

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.