TITLE: Showing that the space $C[0,1]$ with the $L_1$ norm is incomplete
QUESTION [17 upvotes]: Can anyone think of a relatively easy counter example to remember, which demonstrates that the space $C[0,1]$ with the $L_1$ norm is incomplete?
Thanks!

REPLY [2 votes]: Here is an easy example that shows that $C[0,1]$ with the $L^1$ norm is not complete. Let $$f_n(x) = \begin{cases} nx,& 0 \leq x \leq \frac{1}{n} \\ 1,& \frac{1}{n} \leq x \leq 1\end{cases}.$$
It should not be hard to verify that $f_n$ is a cauchy sequence with respect to the $L^1$ norm. Now if there were a function $f$ such that $f_n \rightarrow f$ in the $L^1$ metric, we would necessarily have that $f$ be $1$ for $0 < x \leq 1$ and $f(0) = 0$. But such a function cannot possibly be continuous so that $C[0,1]$ with the $L^1$ norm is not complete.