I came across two definitions of fractional Sobolev spaces $(W^{s,p}(\mathbb{R}^n))$
Def 1: Let $\mathcal{F}(u)$ denote the Fourier transforn of u.
$W^{s,2}(\mathbb{R}^n) = \left\{u\in L^2(\mathbb{R}^n): \{\xi \mapsto (1+|\xi|)^2 \mathcal{F}(u)(\xi)\} \in L^2(\mathbb{R}^n) \right\}$
Def 2: $s\in (0,1)$
$W^{s,p}(\mathbb{R}^n) = \left\{u\in L^p(\mathbb{R}^n): \int\limits_{\mathbb{R}^n} \int\limits_{\mathbb{R}^n}
\frac{|u(x)-u(y)|^p}{|x-y|^{sp+N}} dx dy < \infty \right\}$
How to prove that these two definitions are equivalent for $p=2$ and $s\in (0,1)$?
A clean proof or a reference with detailed proofs is appreciated.
Thanks in advance