Q:

Differentiability of $a:K\to \mathcal{B}(E,F)$?

Assume that $K\subseteq \mathbb{R}$ is an open subset and $E,F$ are Banach spaces.
Denote $\mathcal{B}(E,F)$ to be collection of all bounded linear operators from $E$ into $F.$
Endow $\mathcal{B}(E,F)$ with the strong operator topology.

Question: Suppose that $a:K\to \mathcal{B}(E,F)$ is a bounded linear operator.
  To show that $a$ is differentiable at some $t_0\in K,$ what should we show?

I think we can show the following:
Fix $t_0\in K.$
For $e\in E,$ if $a(\cdot)e:K\to F$ is differentiable at $t_0,$ then we can conclude that $a$ is differentiable at $t_0.$
Am I right?

A:

You're right.
$$\begin{eqnarray} &a(t)\text{ is differentiable at }t=t_0&  \\\iff& \text{strong limit } \lim_{h\to 0} \frac{a(t_0+h)-a(t_0)}{h}&\text{ exists in }\mathcal{B}(E,F). \\\iff&
\forall e \in E, \;\lim_{h\to 0} \left(\frac{a(t_0+h)-a(t_0)}{h}\right)(e)&\text{ exists in }F.\\
\iff&\forall e\in E,\; \lim_{h\to 0} \left(\frac{a(t_0+h)e-a(t_0)e}{h}\right)&\text{ exists in }F.\\
\iff&\forall e\in E,\; t\mapsto a(t)e\text{ is differentiable at }&t=t_0.
\end{eqnarray}$$

