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I need to use the following theorem in an essay I'm writing.

Let $H$ be a real valued Hilbert space and let $(f_m)$ be a sequence of continuously differentiable functions, $f_m:[a,b]\to H$. If $(f_m)$ converges uniformly to $f$ and $(f_m')$ converges uniformly to $g$ on $[a,b]$, then $f$ is differentiable and $$f'=g.$$

Now my supervisor says that the proof requires tools from vector integration theory, but he has forgotten where he has seen it. I am not looking to prove it in the paper, as vector integration theory is way out of the scope of my essay, but I would like a reference to a proof if anyone knows of one.

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    If $f_m'$ converges uniformly to $g$ and $f_m$ converges to $f$, then $f_m(x)=f_m(a)+\int_{a}^{x}f_m'(t)dt$ gives $f(x)=f(a)+\int_{a}^{x}g(t)dt$. You can use vector-valued Riemann integration. – 2017-01-04
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    @TrialAndError Thanks for the succinct outline of the proof. – 2017-01-04

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If I m not mistaken you can find a proof in the book of Amann and Escher Analysis 2. There' s a whole chapter on differentiation in Banach Spaces in it.

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    Page 373 Theorem V.2.8 is exactly what I'm looking for, thanks a lot. – 2017-01-04