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How can I use the Banach-Steinhaus' Uniform boundedness principle in order to prove the following claim:

If $x_n$ is a sequence of complex numbers such that the series $\sum_1^\infty x_n \chi_n$ converges for every sequence $ \chi_n \in l_p $ ($1 \leq p < \infty $ ) , then $x_n \in l_q $ where $ \frac{1}{p} + \frac{1}{q} = 1 $ .

Thanks in advance!

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    The integral version of this was discussed [here](http://math.stackexchange.com/q/61458) (see also the links in the comments there).2012-08-17
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    [This](http://math.stackexchange.com/questions/37647/if-sum-a-n-b-n-infty-for-all-b-n-in-ell2-then-a-n-in-ell2) link, nested in t.b.'s link, directly addresses a particular case of your proposition.2012-08-17
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    Great ! THanks both of you !2012-08-17
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    I'm sorry, but your answer is about Hilbert spaces, and not Banach... How can I fix it ? Thanks2012-08-18

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