  We extend the Global Compactness result by M. Struwe (Math. Z, 1984) to any
fractional Sobolev spaces $\dot{H}^s(\Omega)$ for $0<s<N/2$ and $\Omega \subset
\mathbb{R}^N$ a bounded domain with smooth boundary. The proof is a simple
direct consequence of the so-called Profile Decomposition of P. Gerard (ESAIM:
Control, Optimisation and Calculus of Variations, 1998).
