Let $\dot{H}^{1}(\mathbb{R}^n)$ be the homogeneous Sobolev space formed by completion of $C^{\infty}_{c}(\mathbb{R}^n)$ functions with respect to the norm $||\nabla u||_{L^2}.$ Thus the inner product on $\dot{H}^{1}(\mathbb{R}^n)$ we have, $$\langle u,v\rangle = \int \nabla u \cdot \nabla v$$ for any $u,v\in \dot{H}^{1}(\mathbb{R}^n)$. If we consider the dual space $H^{-1}(\mathbb{R}^n),$ then what will be the inner product on this space?
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