Density in sobolev spaces?

Question

Is the Sobolev space $H_0^1(I)$ dense in $L^2(I)$, where $I\subset\mathbb R$? If so, how do I prove it?

Answer 1

Smooth functions are dense in $L^2$, and you can pick an arbitrarily small (in whatever norm you like) smooth function with specified boundary values on $I$. So if $f_n \to f$, $f \in L^2$, you can replace $f_n$ with $f_n-g_n$, where say $\|g_n\|_{H^1} \leq \varepsilon$. So $f_n - g_n \to f$ in $L^2$, and $f_n-g_n$ is a smooth function that vanishes on the boundary of $I$ hence is in $H^1_0$.