I'm trying to understand the following definition:
which can also be found here on page 136.
Question 1: Closure with respect to what norm? It's not given in the definition.
Question 2: Do I have this right: I can view $C^\infty$ as a dense subspace of $H^k$ via the map (embedding) $f \mapsto (D^\alpha f)_\alpha$ where the tuple $f$ is mapped to consists of all derivatives $D^\alpha f$ such that $|\alpha| \leq k$. Then this is cool because if we have this we can extend any linear operator $T: C^\infty \to C^n$ continuously to all of $H^k$ so that anything we can do to smooth functions we can also do to Sobolev functions. That is, even if the functions don't have a strong $\alpha$-th derivative we can treat them as if they did.
Thanks for your help.