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I have some problems to apply normal convergence of series of functions in any vector space. In fact $(f_{n})$ is a sequence of differentiable functions defined from a topological space $X$ to a normed vector space $Y$, that is normally convergent, I want to ask if the sequence of derivatives $(f_{n}')$ converges normally as well?

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The formulation is a little unclear (should $X$ be a topological vector space?) but in any case the answer is going to be negative. There is never a reason for derivatives to converge, unless we restrict attention to functions that solve some nice elliptic equation and therefore satisfy interior regularity estimates.

A typical counterexample is the sequence $f_n(x)=\frac{1}{\sqrt{n}}\sin nx$ on the real line.

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