So, I've found the result that orthonormal sequences in Hilbert spaces always converge weakly to zero. I've only found wikipedia's "small proof" of this statement, though I have found the statement itself in many places, textbooks and such.(adsbygoogle = window.adsbygoogle || []).push({});

I've come to understand that this property follows from the Bessel inequality, and I've worked out many of the details, so I feel that I understand the Bessel inequality itself quite well. What I don't get is how the inequality gives us the weak convergence - the proof on wikipedia only states that "Therefore, [tex] |\langle e_n, x \rangle |^2 \rightarrow 0 [/tex]" after stating the Bessel inequality. It doesn't make sense to me - how is this information gleaned?

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# Weak convergence of orthonormal sequences in Hilbert space

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