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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.

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

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