  In this note, we establish a strong form of the quantitive Sobolev inequality
in Euclidean space for $p \in (1,n)$. Given any function $u \in \dot
W^{1,p}(\mathbb{R}^n)$, the gap in the Sobolev inequality controls $\| \nabla u
-\nabla v\|_{p}$, where $v$ is an extremal function for the Sobolev inequality.
