Question about definition of Sobolev spaces

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.

Solution 1:

1. We can endow $V$ with the norm $$\lVert (v_j)_{1\leq j\leq K(k)}\rVert_V:=\sqrt{\sum_{j=1}^{K(k)}\lVert v_j\rVert_{L^2(\mathbb T^d)}^2}$$ (it's an usual norm in product of $L^2$ spaces).

2. Yes, by definition $C^{\infty}(\mathbb T^d)$ is dense in its closure. When you say "anything we can do with smooth function, we can do it for Sobolev functions", careful. The property true for smooth functions has to be preserved by taking the limit in $\lVert\cdot\rVert_V$.