I am trying to understand Sobolev's embedding theorem, more precisely to understand when a Sobolev generalized function of infinite order is smooth of some order.
Consider the following statement (from these online notes - pdf file, page 62):
If $f \in H^k (\Omega) = W^{k, 2} (\Omega)$, then $f \in C^s (\overline \Omega)$ for all $s < k - \frac n 2$.
I don't get it. Does this mean that if $k = \infty$ and $s \in \Bbb N$, then every element of $H^\infty (\Omega)$ is smooth of order $s$, and thus it belongs to $C^\infty (\overline \Omega)$, so that all elements of $H^\infty$ are in fact smooth functions? Or is the theorem not valid for $k = \infty$?