Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
add comment

1 Answer

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.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.