Suppose $X$ is a Riemannian manifold. Then we get a Laplace operator on $C^\infty(X)$. In most texts I see the Laplace operator extended to $L^2(X)$, but I don't see how, since it does not seem to be continuous for the $L^2$ norm (There are small functions with big second derivatives, e.g. $f(x) = \epsilon\sin\frac{x}{\epsilon}$ on $[0, 2\pi]$).
Is my example wrong? If not, how do we extend the Laplace operator to $L^2(X)$ in a canonical way, and what good is an extension if it's not continuous?
If I just care about the eigenfunctions of the Laplace operator, do I need to extend it to $L^2$, or do they already occur in $C^\infty$?
There is a Laplace operator on $\Omega^k(X)$. Is there a Hilbert space analogous to $L^2$ to which this Laplace operator extends?