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If $\{x_n\} \subset \ell^1$, then $\sum_{j=1}^\infty x_n(j)y(j)\to 0$ for every $y\in c_0$ iff $\sup_n||x_n||_1< \infty$ and $x_n(j)\to 0$ for $j\geq 1$.

I can proof it by the principle of uniform boundedness; but my question is "why is not it correct for every $y\in \ell^\infty$ (instead of $y\in c_0$)?"

Martin Sleziak
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nika
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    Take $x_n$ to be the standard $n$'th unit vector in $\ell_1$ and $y=(1,1,1,\cdots)$. The reverse implication does not hold here. – David Mitra May 01 '14 at 13:53

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