I am trying to understand the proof for some lemma in a book, but the part the authors label as trivial is not trivial to me at all. Any help is appreciated. Here is the trivial part of the lemma:
$X$ is a Banach space, $A$ is a function from a compact set $K \subset \mathbb R$ into $L(X)$.
If $A$ is continuous for the topology of uniform convergence on compact subsets of $X$ then $A$ is continuous for the strong operator topology.