I would like to show the following inequality regarding the dual semigroup of a semigroup of linear operators (the one at the end of the image). The screenshot comes from the book One-paramter Semigroups of Positive Operators of Arendt et al.
I have troubles understanding why $$ ||f||\leq M \sup\{\langle f, \phi \rangle \mid \phi\in E^*, ||\phi||\leq 1\}. $$ They write that one should consider $\int_0^t{T'_s \phi}dt$ understood in a weak sense, i.e. as a map $f\mapsto \int_0^t{T'_s \phi(f)}dt$. So first of all I think one needs to consider $\psi_t=\frac{1}{t}\int_0^t{T'_s \phi}dt$ since $\frac{1}{t}\int_0^t{T_tf}dt\rightarrow f$ strongly. So I tried the following $$ ||f||\leq||\int_0^t{T_tf-f}dt||+||\int_0^t{T_tf}dt||. $$
The first term on the righthandside above goes to $0$ hence is smaller than an $\varepsilon$. However hard we try we cannot estimate the second term with $||\psi_t||$.
Another idea I had was to use Hahn-Banach to find a $\phi$ with $||\phi||=||f||$. Since we are only interested in $f$ with $||f||\leq 1$ this shouldn't be a problem. But we don't really want the $||\cdot||$-norm of $E'$ on the right-hand side.
Let $\psi_t$ be as in the question with $\|\phi\|\leq 1$. We have $\psi_t\in E^\ast$ and $$ \|\psi_t\|\leq \frac 1 t\int_0^t \|T_s'\phi\|\,ds\leq \frac 1 t\int_0^t\|T_s\|\,ds. $$ In particular, for every $\epsilon>0$ there exists $T\geq 0$ such that $\|\psi_t\|\leq M+\epsilon$ for $t<T$.
Thus $$ \phi(f)=\lim_{t\searrow 0}\psi_t(f)\leq (M+\epsilon)\sup\{\psi(f)\mid\psi\in E^\ast,\,\|\psi\|\leq 1\} $$ for every $\epsilon>0$, which implies $$ \|f\|=\sup\{\phi(f)\mid \phi\in E',\|\phi\|\leq1\}\leq M\sup\{\psi(f)\mid\psi\in E^\ast,\,\|\psi\|\leq 1\}. $$