In Gilbarg/Trudinger chapter 9.4, in the proof of Calderon-Zygmund inequality for $p=2$ case, the auther uses the approximation argument as follows:(here $w=Nf$(Newtonian potential))
Namely, first prove bound for compactly supported smooth functions, then approximate given $L^2$ function by such functions(in $L^p$ sense).
By lemma 7.12, this proves that $w_m=Nf_m \rightarrow w=Nf$ in $L^2$. Furthermore by (9.28), ${w_m}$ forms a bounded sequence in $W^{2,2}$, hence it contains a subsequence which converges weakly in $W^{2,2}$ and strongly converges in $L^2$.
Hence I concluded that (by replacing to subsequence) $w_m$ weakly converges to $w$ in $W^{2,2}$, which implies that $w$ is in $W^{2,2}$, and $$ \|w\| _{W^{2,2}} \leq \liminf_{m \to \infty} \|w_m\| _{W^{2,2}}\leq\|f\| _{L^2} $$
But this is not enough to prove the equality in (9.28) in general case and in fact the auther claims that $w_m$ is actually convergeing to $w$ strongly in $W^{2,2}$.
Not only in this place but also at the end of the proof of lemma 9.12, for $u \in W^{1,1}_0$, mollified functions $u_h \in C^{\infty}_0$ claimed to converge to $u$ in $W^{2,p}$ where $u_h$ are showen to be uniformly bounded in $W^{2,p}$.
So I guess that there is some argument justifying these kind of strong convergence in Sobolev spaces, but I cannot find it by myself. How can I prove such convergence in these cases? Is there any general argument applicable in such situations? Also, is my argument about weak convergence to $w$ correct? Any help would be appreciated.
The argument shows that $N|_{C_0^{\infty}(\Omega)}$ is a bounded linear operator from $(C_0^{\infty}(\Omega), ||\cdot||_{L^2})$ to $W^{2,2}(\Omega)$. Hence, it can be uniquely extended to a bounded linear operator from $L^2(\Omega)$ to $W^{2,2}(\Omega)$. This extension is also bounded from $L^2(\Omega)$ into $L^2(\Omega)$, and thus, must coincide with $N$. Therefore, $N:L^2(\Omega) \to W^{2,2}(\Omega)$ is bounded. We infer that $Nf_m \to Nf = w$ in $W^{2,2}(\Omega)$.