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Suppose we have a sequence in a Hilbert Space, which converges weakly to some limit, and it has a subsequence which converges strongly to that same limit. Does this imply the sequence is strongly convergent? I would appreciate if someone could show me a proof / counterexample. Thank you :)

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    $\begingroup$ No. Say $x_n\to0$ weakly but $||x_n||\not\to0$. Consider the sequence $x_1,0,x_2,0,\dots$. $\endgroup$ – David C. Ullrich Nov 18 '17 at 18:13

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