Weak differentiability

Question

Let $v\in L^{2}(\Omega)$ and for every $\phi \in C^{\infty}_{c}(\Omega)$

$|\int_{\Omega}v\phi'dx|\leq C\|\phi\|_{L^{2}(\Omega)}$

holds. Does it imply that $v$ is weak differentiable?

Answer 1

Yes because it is telling you that the functional $$T(\phi)=-\int_\Omega v\phi^\prime dx,\quad C^1(\Omega)\cap L^2(\Omega)$$ is linear and continuous. So you can first apply Hahn-Banach theorem to extend it to a linear continuous functional $T:L^2(\Omega)\to\mathbb{R}$ and then use Riesz representation theorem in $L^2$ to write $T(\phi)=\int_\Omega g\phi\,dx$ for some $g\in L^2(\Omega)$. The function $g$ is the weak derivative of $v$