Closure of a nontrivial normed vector subspace that is equal to the whole space

Question

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 $V$?

Answer 1

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 $L^2[0,1]$.

Answer 2

Of course, we have to look at infinite dimensional vector spaces, because in the finite dimensional case, any subspace is finite dimensional hence closed, and cannot be dense if it's strict.

You can take the probability space $([0,1],\mathcal B([0,1]),\lambda)$, $F:=L^2[0,1]$ and $E=L^1[0,1]$. Then $F\subset E$ and $F$ is a subspace of $E$. It's a strict subspace (take $f(x)=\frac 1{\sqrt x}$) and dense: take $f\in L^1$, $f_n:=f\chi_{|f|\leq n}$, then $$\lVert f-f_n\rVert_1=\int_{[0,1]}|f(x)|\chi_{|f(x)|>n}d\lambda=\sum_{k\geq n+1} \int_{[0,1]}|f(x)|\chi_{k\leq |f|<k+1}d\lambda\\\leq \sum_{k\geq n+1}k\mu(A_k)+ \sum_{k\geq n+1}\mu(A_k),$$ where $A_k=\{k\leq |f|<k+1\}$. Since $f\in L^1$, the series $\sum_{k\geq 1}k\mu(A_k)$ and is convergent so $\lVert f-f_n\rVert_1\to 0$.