Sequence of differentiable Hilbert Space valued function-Reference request.

Question

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.

Answer 1

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.