Let $$ P^0:L^2(\Omega)\to L^2(\Omega)\\ P^1:(L^2(\Omega))^3\to (L^2(\Omega))^3 $$ be bounded operators. In particular, the images of $P^0$ and $P^1$ are subsets of some suitable spline spaces.
We want to prove that $$ \text{grad}\,(P^0 f)=P^1(\text{grad}\, f)\qquad \forall\,\, f\in H^1(\Omega). $$ I proved the statement for every $f\in C^\infty_c(\Omega)$. Then, I considered a convergent sequence $\{f_n\}_n$ such that $f_n\to f \in H^1(\Omega)$, where the convergence is wrt to $\|\cdot\|_{H^1(\Omega)}$.
I can't write down the density argument explicitly. Any help?