Closure of a nontrivial normed vector subspace that is equal to the whole space Can you show me an example of a normed vector subspace $S$ strictly included in a normed vector space $V$ whose closure is equal to the whole $V$?

Expanding on David's example: the span of the canonical basis in $\ell^p(\mathbb{N})$, $1\leq p <\infty$.

Another example: $C[0,1]$ in $L^2[0,1]$.