I am reading this script: http://www.hairer.org/notes/SPDEs.pdf about stochastic partial differential equations. I have a problem with the footnote on page 46.
$L$ is the generator of some $C_0$ semigroup on a Banach space $B$ (one can assume separability here but I think that doesn't play a role).
From that we know that $L$ is a closed operator and the domain $\mathcal{D}(L)$ is dense in $B$.
In the footnote he wants to show that some subspace of $B^*$ already seperates points in $B$ and what he needs to show is that the range of the adjoint operator $L^*: B^* \to B^*$ is already all of $B^*$.
The first thing he does is assume that the range of $L$ is already all of $B$ and says we can do that without loss of generality. And here I am lost already. I don't even recall that the range should be dense in $B$ and even if, I don't see how you can make such a strong assumption w.l.o.g..
(Actually I don't understand his argumentation afterwards too and how exactly he applies the closed graph theorem, but if I assume that the range $\mathcal{R}(L)$ is already $B$ from my view one could just apply the closed range theorem and its corollary wikipedia link)