# 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 $V$?

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The linear span (finite linear combinations) of the standard unit vector in $\ell_1$. – David Mitra May 17 '12 at 2:23

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 $L^2[0,1]$.
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 $\lVert f-f_n\rVert_1\to 0$.