How to prove the following is semi-norm $$[u]_{s,p}=\Bigg(\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}dxdy\Bigg)^{1/p}$$ where $\Omega$ open set in $\mathbb R^n$, $1\leq p<\infty$, $0<s<1$.
As a semi-norm satisfying triangle inequality and homogeneous. I am facing problem in triangle inequality.
Obviously $[.]_{s,p}$ is homogeneous. Define for $u:\Omega\rightarrow\mathbb{R}$ the function $\phi_u:\Omega\times\Omega\rightarrow\mathbb{R}$ by: $$\phi_u(x,y)=\frac{|u(x)-u(y)|}{|x-y|^{\frac{n}{p}+s}}.$$ Then $||\phi_u||_{L^p(\Omega\times\Omega)}=[u]_{s,p}$.By the triangle inequality on $\mathbb{R}$
$$\phi_{u+v}(x,y)=\frac{|(u+v)(x)-(u+v)(y)|}{|x-y|^{\frac{n}{p}+s}}\leq \frac{|u(x)-u(y)|}{|x-y|^{\frac{n}{p}+s}}+\frac{|v(x)-v(y)|}{|x-y|^{\frac{n}{p}+s}}$$ for all $x,y\in\Omega$ and thus $$||\phi_{u+v}||_{L^p(\Omega\times\Omega)}\leq ||\phi_u+\phi_v||_{L^p(\Omega\times\Omega)}\leq||\phi_u||_{L^p(\Omega\times\Omega)}+||\phi_v||_{L^p(\Omega\times\Omega)}$$, by the Minkowski-inequality and that yields $$[u+v]_{s,p}\leq[u]_{s,p}+[v]_{s,p}$$ which makes the Gagliardo semi-norm a real semi-norm but no norm since all constant functions have Gagliardo semi norm zero.