Suppose $\Omega$ is a bounded open subset of $\mathbb{R}^n$ with a "nice" boundary. We have the Sobolev spaces $W^{k,2}(\Omega)$, which are all contained within each other: $W^{m,2}(\Omega)\subset W^{k,2}(\Omega)$ if $m>k$. What is the intersection $$\bigcap_{k}W^{k,2}(\Omega)\,?$$ Could it be $C^\infty(\bar{\Omega})$, the infinitely differentiable functions on the closure of $\Omega$?
(In the case I'm interested in, the boundary of $\Omega$ is just a bunch of hyperplanes and their intersections. In fact, $\Omega$ is just an $n$-simplex.)
And could I deduce from this that $C^\infty(\bar{\Omega})$ is a nuclear Fréchet space?
