Let $\{f_{n}\}$ be a sequence of nonzero continuous functions on $\mathbb{R}$, which is uniformly bounded, uniformly Lipschitz on $\mathbb{R}$, and the derivative sequence $\{f_{n}'\}$ is also uniformly Lipschitz on $\mathbb{R}$, and $f_{n}\in L^{2}(\mathbb{R})$ for all $n$. If $\{f_{n}\}$ converges uniformly on any closed interval $I\subset \mathbb{R}$ to a continuous function $f$, does this imply that $f\in L^{2}(\mathbb{R})$. If not, what condition(s) the sequence $\{f_{n}\}$ must have to get such result?
Sequence of square integrable functions
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real-analysis
sequences-and-series
convergence-divergence
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0Just curios: are you another incarnation of berry who posted this question: http://math.stackexchange.com/questions/1595$0$7/sequence-of-lipschitz-functions/15952$0$#$1$5952$0$ ? Or are you just attending the same class? – 2012-06-18
1 Answers
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You need a bound on the $L^2$ norms of the $f_n$. Since you know that your sequence converges pointwise (even better) the same is true for $|f_n|^2$. You can then use Fatou's lemma applied to $|f_n|^2, |f|^2$ to conclude that the limit is in $L^2$.
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0if the sequence of $L^2$ norms is unbounded you are probably out of luck. I do admit that I do not have a counterexample right now. If you know that the sequence of $L^2$ -norms is unbounded you seem to have additional information about the $f_n$. Why don't you add this to your question, this may make it easier to provide an answer. – 2012-06-18