Weakly convergent but not strongly convergent sequence in $L^2(\Bbb R^n)$.

Question

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.