The Bessel Potential Space is defined for $s\in\mathbb{R}$ as,

$H^s(\mathbb{R}^d) = \{f\in L_2(\mathbb{R}^n) : (1+|\cdot|)^{s/2}\hat{f}(\cdot)\in L_2(\mathbb{R}^n)\}. $

This defines a Hilbert space such that for any $f,g\in H^s(\mathbb{R}^n)$,

$ \langle f, g\rangle = \int_{\mathbb{R}^n} \hat{f}(\omega)\overline{\hat{g}(\omega)} (1+|\omega|)^{s}d\omega. $

For any open set $\Omega\subset\mathbb{R}^n$ we have $H^s(\Omega)$ being the set of restrictions with norm,

$ \left\|f\right\|_{H^s(\Omega)} = $

$ \inf_{g\in H^s(\mathbb{R}^n)}\{\left\|g\right\|_{H^s(\mathbb{R}^n)} : g|\Omega=f \} $

Does this definition of the norm ensure we have the following: Given an open set $\Omega\subset \mathbb{R}^n$ and open sets $\Omega_1, \Omega_2\subset \mathbb{R}^n$ such that $\Omega = \Omega_1\cup \Omega_2$ and $f\in H^s(\Omega)$,

$ \left\|f\right\|^2_{H^s(\Omega)}\leq \left\|f\right\|^2_{H^s(\Omega_1)} + \left\|f\right\|^2_{H^s(\Omega_2)}. $