The following is an exercise from Dunford & Schwartz (1958) page 653.
Let $X$ be a Banach space and $T_t:X \to X$ be a semigroup of bounded operators indexes by $\mathbb{R}_{\geq0}$. Suppose it's continuous w.r.t the weak operator topology, i.e.
$$\forall \; x \in X, \; y^* \in X^*, \; t \to y^*(T_t(x)) \; cont.$$
Prove that it's strongly continuous, i.e. $t \to T_tx$ is norm continuous for all $x\in X$.
What I Tried:
Obviously we can only prove continuity at $t=0$, and one can show using applications of the uniform boundness principle that $\Vert T_t \Vert \leq Me^{ct}$. And From here we can reuce to the case of a semigroup of contractions on a Banach Space. But In general, I am not sure that if $x_n \overset{w}{\to} x$ weakly and $\underset{n\to \infty}{limsup}\Vert x_n \Vert \leq \Vert x \Vert$ then there is norm convergence.
Any help is appreciated.
Best Answer
This result is a theorem (with proof) in Engel's book.