Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 5 / 4 / b / Solution

For every compactly supported smooth test function $\varphi$, integration by parts in the definition of the weak derivative givesBoth $\varphi$ and ${\partial }_{\ell }\varphi$ belong to $L^2({\mathbb{R}}^n)$, so the assumed weak limits implyThis is exactly the distributional characterization of ${\partial }_{\ell }g=h_{\ell }$. Since $g,h_1,\dots ,h_n$ are all in $L^2$, the defining condition of the H1 space is satisfied. ThusFor complex functions the same argument uses conjugated test functions in the $L^2$ pairing. No prior identification of an $H^1$ weak limit is needed.