can we say that $f_j\to 0$ weakly.

Question

for $j\in\Bbb{N}$, we put $f_j(x,y)=\sqrt{j}\exp(-j(x^2+y^2))$. so $f\in(L^2(\Bbb{R}^2),\|\cdot\|_2)\cap C^{\infty}(\Bbb{R}^2)$ with $||f_j||_2=const$ for all $j$.

We have $f_j\to 0$ a.e;

I want to show there is no subsequence of $(f_j)$ which converge in $(L^2(\Bbb{R}^2),\|\cdot\|_2)$. Indeed, if $f_{j_k}\to f$ in $(L^2(\Bbb{R}^2),\|\cdot\|_2)$, then there is a subsequence $f_{j_{k_p}}$ of $f_{j_k}$ which converge a.e to $f$ and hence $f=0$. But $\lim_p (||f_{j_{k_p}}||- ||f||)=0$ so $\lim_p||f_{j_{k_p}}||=0$ absurd.

My question: can we say that $f_j\to 0$ weakly.