Basic properties of uniform limits in Banach spaces

Question

Where can I find infos (books, keywords, online materials, etc.) about when the uniform limit of a sequence of continuously differentiable functions $f_n:U\subseteq E\rightarrow F$ between arbitrary Banach spaces is itself continuously differentiable?

Thanks

Answer 1

The basic theorem in this regard is that the conclusion holds so long as the sequence of derivatives $df_n$ converges locally uniformly. See 8.6.3 of Foundations of Modern Analysis by Dieudonné.