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Let us consider the sequence $f_n(x,y)=\exp[-\frac{1}{|\sin(n)|} (x^2+y^2)] \in L^2(\mathbb{R}^2)$ with $n\in \mathbb{N}^*$.

Can be extracted from it a subsequence converges in $L^2(\mathbb{R}^2)$?

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    Can be extracted from it a subsequence converges in $L^2$ – 2017-01-22
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    Think about extracting a subsequence of $(\sin n)$ which converges to 1. – 2017-01-22
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    thank you,one can find a such sequence . – 2017-01-22

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You can calculate: $$\|f_n\|_{L^2}≤\|f_n\|_{L^1}=\pi|\sin(n)|$$

Since there is a subsequence of $\sin(n)$ that converges to zero you have a subsequence of $f_n$ so that the norm converges to zero.

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    You are right. in fact I search a bounded radial sequence in $L^2$ from which we can not extract a convergent subset. – 2017-01-22
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    @FadilKikawi radial meaning? – 2017-01-22
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    I had a brain-fart, sorry. The correct sequence is $f_n(x,y)=\begin{cases}1&x^2+y^2\in[\sqrt n,\sqrt{n+1}]\\0&\text{otherwise}\end{cases}$. Here the norm is $\sqrt{\pi}$. – 2017-01-22