The Sobolev spaces with fractional order $H^s(\Omega)$ ($s>0,n\in\mathbb{N}, \Omega\subseteq \mathbb{R}^n$ open) in Hitchhikers guide is defined wrt. the norm
$$\|u\|^2_{s}:= \|u\|_{H^m(\Omega)}^2+\sum\limits_{|\alpha|=m}\Big(\|D^\alpha u\|_{L^2(\Omega)}^2+ \int\limits_{\Omega \times \Omega} \frac{|D^\alpha [u(x)-u(y)]|^2}{|x-y|^{n+2\sigma}} d(x,y)\Big).$$
But it seems that the norm could also obviously be defined by
$$\|u\|^2_{s}:= \|u\|_{H^m(\Omega)}^2+\sum\limits_{|\alpha|=m}\int\limits_{\Omega \times \Omega} \frac{|D^\alpha [u(x)-u(y)]|^2}{|x-y|^{n+2\sigma}} d(x,y).$$
What is the purpose of the additional term $\|D^\alpha u\|_{L^2(\Omega)}$?