Operators, Riesz

Question

Suppose that $\Omega\subset\mathbb{R}^n$ is open, $k\mapsto f_k\in H^{1,2}(\Omega)$ and $f\in L^2(\Omega)$ such that $f_k\rightarrow f$ in $L^2(\Omega)$ and $||\partial_if_k||_{L^2(\Omega)}\le K\ \ \forall i\in\{1,\ldots ,n\}$.

1. Let $T_i:\mathcal{C}^{\infty}_c(\Omega)\rightarrow \mathbb{R},\quad T_i:\phi\mapsto \int_{\Omega} f\partial_i\phi \ d\lambda_n$ Show that $|T_i\phi|\le K||\phi||_{L^2(\Omega)}\ \forall \phi\in \mathcal{C}^{\infty}_c(\Omega)$.

2. Show that $f\in H^{1,2}(\Omega)$ and $||\partial_i f||_{L^2(\Omega)}\le K \ \forall i\in\{1,\ldots ,n\}$.