I am currently reading up on Stone's Theorem on one-parameter unitary groups and I came across following claim in Wikipedia (https://en.wikipedia.org/wiki/Stone%27s_theorem_on_one-parameter_unitary_groups#cite_note-1):
Consider the one-parameter family $(U_t)_{t \in \mathbb{R}}$ of unitary operators defined by $U_{t}:=e^{itA}$, for all $t \in \mathbb{R}$, where $A$ is a self-adjoint operator on a Hilbert space $\mathcal{H}$. Then, $(U_t)_{t \in \mathbb{R}}$ is a strongly continuous one-parameter group. Moreover, A will be bounded if and only if the operator-valued mapping $t \mapsto U_{t}$ is continuous with the operator norm.
I can see why $(U_t)_{t \in \mathbb{R}}$ strongly continuous. However, I have been struggling to prove the second claim. That is, A will be bounded if and only if the operator-valued mapping $t \mapsto U_{t}$ is continuous with respect to the operator norm.
Any hint or help will be greatly appreciated. Thanks.