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.
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It is $f\in\mathcal{L}(E,F)$. Sorry about the miswriting. – Ray Romero Mar 21 '20 at 21:02
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Can you cite the definition of differentiation in this context? And where are you stuck? – Berci Mar 21 '20 at 21:15
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What is the best linear approximation to a linear map? Itself – Ivo Terek Mar 21 '20 at 21:32
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Notice that, since $f$ is linear:
$$ f(a+h) - f(a) = f(h) $$
hence the rest $R(h) = 0$, what gives the result.
Danilo Gregorin Afonso
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Why did you unaccepted the answer? Do you need clarifications? – Danilo Gregorin Afonso Apr 05 '20 at 22:32