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Let $\{a_{n,m}\}_{n,m \in \Bbb N}$ be a double complex sequence such that:

$$\lim_{m \to \infty} \sum_{n=1}^\infty |a_{n,m} |^2 =0$$

I would like to know if the following relation is true:

$$\lim_{m \to \infty} \sum_{n=1}^\infty a_{n,m} =0$$

Thanks

1 Answers 1

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No thats not true. Take $a_{n,m}=\frac{1}{nm}$.

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    Now it does. I corrected the typo. Thank you! – 2017-02-03
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    ok thanks so much, but if i know that it is convergent, in this case is it true? – 2017-02-03
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    What do you mean by _i know that it is convergent_? – 2017-02-03
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    i mean that $\lim_{m \to \infty} \sum_{n=1}^\infty a_{n,m} $ is convergent – 2017-02-03
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    No, why should $\lim_m \sum_n a_{n,m} <\infty $ imply $\lim_m \sum_n a_{n,m} =0$ or what do you mean? – 2017-02-03
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    i mean if $\lim_{m \to \infty} \sum_{n=1}^\infty \lvert a_{n,m} \rvert ^2 =0$ and $\lim_{m \to \infty} \sum_{n=1}^\infty a_{n,m}$ is convergent => $\lim_{m \to \infty} \sum_{n=1}^\infty a_{n,m} =0$ – 2017-02-03
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    But this is exactly your question and I already gave you a counterexample. – 2017-02-03
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    excuse me, your counterexample is for $\lim_{m \to \infty} \sum_{n=1}^\infty a_{n,m} $ not convergent – 2017-02-03