I want to calulate the range of an operator $S$ which maps $L^{2}$ into $H^{1}(\Omega)$?

Question

I have an equation $s(\lambda,\mu)=l(\mu)$, where s(.,.) is a symetric positive definite bilinear form in $L^{2}(\Omega)$, and $l(.)$ is in $H^{1}(\Omega)$. I want to show that the range of the operator $S$ associated to $s(.,.)$ is $H^{1}(\Omega)$. Thank you