I am looking at the Sobolev space
$W^s=\{ f \in L^2(\Omega): \partial^{\alpha}f \in L^2(\Omega) \ \forall \ |\alpha|\le s \} $
for some $\Omega \in \mathbb{R}^n$.
How can one proof that the norms:
$$||f||=\sum_{|\alpha| \le s} ||\partial^{\alpha}f||_{L^2}$$
and
$$|||f|||=||\hat{f}(k)(1+|k|^2)^{\frac{s}{2}}||_{L^2}$$
are equivalent?
Here $\hat{f}$ denotes the Fourier transform of $f$.
I know I can write $\|f\|=∑_{|α|≤s|}k^α\|\hat f\|_{L^2}.$ Then I think one can show that $\sum_{|\alpha| \le s} |k^{\alpha}\hat{f}| \le |(1+|k|^2)^{s/2}\hat{f}|,$ since the expansion of the brackets contains all the terms $|α|≤s.$