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Consider a sequence $(x_n)_n$ in Hilbert space $H$ such that $\langle x_m,x_n\rangle=\delta_{mn}$ where $\delta_{mn}$ equals one if $m = n$ and $C$ otherwise. Prove that $(x_n)_n$ is a weakly convergent sequence.

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    One could use [Bessel's inequality](http://en.wikipedia.org/wiki/Bessel%27s_inequality).2012-12-09
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    Bessel's inequality could be applied to an orthonormal sequence only.2012-12-09
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    Oops, I misread that "$C$"...2012-12-09
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    If the problem is correct, we necessarily have $C=0$, otherwise we have $\langle x_m,x_1\rangle =C$ for $m\geqslant 2$.2012-12-09
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    Could you explain this in detail?2012-12-09
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    I see that your consequences $\langle x_m,x_1\rangle =C$ for $m\geq 2$ is the assumption of problem.2012-12-09

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