Let $U$ be a open set of $\mathbb{R}^n$, C(U) is all continuous functions on U, for example C(0,1), when $U=(0,1)$. And $L^1(U)$ is lp-space where $p=1$.
It was said that $L^1(U)$ is the completion of of $C(U)$ by the norm $\|\cdot\|_{L^1}$. Namely $$ L^1(U)=\overline{C(U)}. $$
But I feel confuse, the function in $C(U)$ my be not integrable, $1/x$ in $(0,1)$ for example. But from $\overline{C(U)}=L^1(U)$ we can infer that $C(U)\subset L^1(U)$. This is a contradiction.