I have some questions while reading "An introduction to Complex Analysis in Several variables, Las Hörmander(3rd Edition). The part what I do not understand is following here.
Let $\varphi$ be a function in $\mathbb{C}^2$ and $L^2_{(p,q)}(\Omega, \varphi)$ be the space of all (equivalent class of) forms of type (p,q) such taht the coefficienets are measurable in any local system and $|| f||_{\varphi}^2$ = $\int|f|^2e^{-\varphi}dV < \infty$ The operator $\bar\partial$ is linear, closed, densely defined operators and T,S are given
T : $L^2_{(p,q)}(\Omega, \varphi)$ $\to$ $L^2_{(p,q+1)}(\Omega, \varphi)$, S : $L^2_{(p,q+1)}(\Omega, \varphi)$ $\to$ $L^2_{(p,q+2)}(\Omega, \varphi)$
Then,
Lemma 5.2.1. $D_{(p,q+1)}(\Omega )$ is dense in $D_{T^*} \cap D_{S}$ for the graph norm $ f \to$ $|| f||_{\varphi}+ ||T^*f||_{\varphi}+||Sf||_{\varphi}$
(Notation : $D_{(p,q+1)}(\Omega )$ = $\{f \in L^2_{(p,q+1)}(\Omega, \varphi) : f \in C_0^{\infty}(\Omega)\} $ Where $C_0^{\infty}(\Omega)$ is smooth function which vanishs outside compact subset of $\Omega$ . $D_{T^*},D_{S}$ are the domain of the linear operator $T^{*},S$.)
And I will attach some shots on the textbook while asking questions.
Question1 : I do not understand the reason $\eta_{\nu}f \to f $ in the graph norm if $f \in D_{T^*}\cap D_{S}$. More presciely, in (converging) in the graph norm.
Question2 : Frankly speaking, I do not understand the gist of the proof. The aim for proof is the set $D_{(p,q+1)}(\Omega )$ is dense subset in $ D_{T^*}\cap D_{S}$. In topological sense, it means $cl(D_{(p,q+1)}(\Omega ))= D_{T^*}\cap D_{S} $ (or equivanlce relation of dense subset in ...) However, the author refers (as seeing the textbook in the 2nd shot),
(♠) It is therefore sufficient to approximate elements in $D_{T^*}\cap D_{S} $ which have compact support, and by means of a partition of unity we reduce the proof to the case when the support lies in a coordinate patch. In that case we can use the following classical lemma of Frieddrichs(Lemma 5.2.2).
Anyway, the author proves the Lemma5.2.2 and then go to the remaining travel
However, I don't know why it is sufficient merely to check (♠) in order to verify the Lemma 5.2.1, $D_{(p,q+1)}(\Omega )$ is a dense subset in $D_{T^*}\cap D_{S}$.(Concretly, I do not catch the relation between the aim for the Lemma 5.2.1 and (♠))