Let $E, F$ normed spaces and $f\in\mathcal{L}(E,F)$. Prove that $\forall a\in E: f'(a)=f$

Question

I have tried to prove it by the definition of differentiability, but I have gotten nowhere. I am not very clear how to proceed. I would really appreciate your help.

Answer 1

Notice that, since $f$ is linear:

$$ f(a+h) - f(a) = f(h) $$

hence the rest $R(h) = 0$, what gives the result.