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I have a sequence of numbers $x_n$ that satisfy that for every $y_n \in c_0$ (when $c_0$ is a Banach space of all the complex sequences that satisfy $\lim_{n\rightarrow \infty }{a_n} =0$ ) the series $\sum_1^\infty{x_ny_n} $ convergence .How do I show that $x_n \in l^1 $ ?

Maybe using uniform boundedness principle will help me ?

Thanks for your help.

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    $\begingroup$ I'd think that Hugo Steinhaus would know how to use the uniform boundedness principle. $\endgroup$
    – Asaf Karagila
    Mar 2 '13 at 21:21
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    $\begingroup$ I hope he will not be mad at me I made him look like an idiot $\endgroup$ Mar 2 '13 at 21:34
  • $\begingroup$ $\sum_{1}^{\infty}x_ny_n$ converges, here $y_n\in c_0$ and $x_n$ is just a number? $\endgroup$
    – Coiacy
    Mar 4 '13 at 18:33
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For each integer $N$, consider $T_N\colon c_0\to \Bbb R$ given by $$T_N(y):=\sum_{j=1}^Nx_jy_j.$$

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