A subspace of $H^1(0,\infty)$

Question

I am just wondering why a space like $H_0^1(0,\infty)=\{f\in H^1(0,\infty):f(0)=0\}$ is dense in $L^2(0,\infty)$ where $H^1$ is the Sobolev space?

Thanks in advance.

Math

Answer 1

$C^{\infty }$ functions with compact support in $(0,\infty )$ are dense in $L^{2}(0,\infty )$. Such functions belong to $H^{1}_0(0,\infty )$.