I have read two definitions of Sobolev spaces.
Definition 1: We let $\lambda$ denote $\lambda^s(\xi)=(1+|\xi|^2)^\frac{s}{2}$ for $s \in \Bbb R$, $\xi \in \Bbb R^n$. We say that $u \in H^s$, if $u \in S'$ and $$||\lambda^s \hat{u} ||_2= (2 \pi)^{-n} \int (1 + |\xi|^2 )^s |\hat{u}(\xi)|^2 \, d \xi < \infty$$ under the identification of $L^2$ in $ S'$. $S'$ is the dual of the Schawrtz Space on $\Bbb R^n$, known as temperate distribution , with respect to the norm on Schawrtz space.
What differentiates this definition to the second definition:
Definition 2: On the Schwartz space, the Sobolev space $W^s$ is the completion of $S$ with respect to the $s$-norm. $$||u||_s^2 := (2 \pi)^{-n} \int (1 + |\xi|^2 )^s |\hat{u}(\xi)|^2 \, d \xi $$
Is that the first definition is working with topologicial dual on Schwartz space whilst latter is working directly with Schwartz space.
Are these the same? Why do we have a $H$ snd a $W$?
Sources: First definition is from Xavier's, Introudction to Pseudodifferential Manifolds, Second Definition is From Ebert's notes. I am quite confused, as both of these definitions do not appear in Sobolev spaces.
As suggested by user Rhys:
So we have a map $S \rightarrow S'$, given $$\phi \mapsto u_\phi= \left( \psi \mapsto \int \psi \bar{\phi} \right)$$ $$||\hat{u}_\phi||_2 = ||u_\phi||$$ by construction. It suffices to show $S$ in the $||\cdot||_2$ norm is
(i) Dense in $H^s$
(ii) and $H^s$ is complete.
Are these true?