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I have seen an example of functions that are weakly convergent but not strongly convergent in $L^2([0,2\pi])$, which is $f_k(x) = \sin(kx)$, but I'm unsure that this example works in $L^2(\Bbb R^n)$, mostly because I don't even think $\sin(x)$ is in $L^2(\Bbb R^n)$. Try as I might, I can't think of an example that works over all $\Bbb R^n$, with the exception of maybe $\sin(kx)\cdot \chi_{[0, 2\pi]}$, but I'm hoping to find a function that is substantially different from this example.

Alex M.
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poppy3345
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    You'll find here an essentially different example in $L^2(R)$: http://math.stackexchange.com/questions/1431026/does-the-sequence-sqrtn-cdot-1-0-1-n-n-converge-weakly-in-l2?rq=1 – uniquesolution Sep 16 '15 at 23:43
  • @uniquesolution ah, got it. Would you mark this question as a duplicate? I didn't see that one before I created this one! – poppy3345 Sep 17 '15 at 00:07
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    Every orthonormal basis gives an example of a sequence that converges weakly but not strongly to 0. – Urgje Sep 17 '15 at 11:15

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