metric space incomplete

Question

Good day, please how show that C[a,b] with norm $\left \| f \right \|=(\int_{a}^{b}f^{p}(x)dx)^{\frac{1}{p}}$ is not complete, $p\geq 1$. Can I do it with $g_{n}=(\int_{a}^{t}f_{n}^{p}dt)^{\frac{1}{p}}$ adecuate?

Answer 1

I'll give you a hint for the case $[a,b]=[-1,1]$. Consider $g_n\in C[-1,1]$ such that $g_n(t)=nt$ for $t\in [-1/n,1/n]$ and $g_n(t)=-1$ on $[-1,-1/n)$ and $g_n(t)=1$ on $(1/n,1]$. Show that
1). $\{g_n\}$ constitute a Cauchy sequence as to the $p$-norm.
2). If $g_n\to g\in C[-1,1]$ in the $p$-norm, then there is no way to define the value of $g(0)$.