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Let $(a_{n})$ be a sequence of complex numbers such that for every $(b_{n})\in \ell^{2}$the series $\sum_{1}^{\infty}a_{n}b_{n}$ converges. Prove that $(a_{n})\in \ell^{2}.$

What I've tried so far is Let $T(b)=\sum_{n=1}^{\infty}a_{n}b_{n}$ where $b=(b_{n})$ then $T$ is a linear operator on $\ell^{2}$. So if I can somehow show that $T$ is continuous then I can apply Riez representation theorem for Hilbert spaces can conclude that $(a_{n})\in \ell^{2}$. I'm thinking maybe something along the line of Principle of uniform boundedness but I'm not getting anywhere with that.


marked as duplicate by Davide Giraudo, Amzoti, user63181, Brian M. Scott, Dominic Michaelis Apr 12 '13 at 21:53

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