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