Bounded seqeunce in $L^2(\Bbb R^d)$ has a weakly convergent subsequence

Let $\{f_n\}$ be a bounded sequence in $L^2=L^2(\Bbb R^d)$, i.e. $\sup_n ||f_n||_{L^2}<\infty$. I am trying to show that it is weakly convergent in the sense that there is $f\in L^2$ and a subsequence $\{f_{n_k}\}$ such that for every $g\in L^2$, $$\lim_{k\to \infty}\int f_{n_k}g =\int fg,$$by using the Riesz representation theorem.

I want to try to mimic the proof here: Bounded sequence in a Hilbert space has weakly convergent subsequence, but I have some questions:

1. Does $L^2(\Bbb R^d)$ has a "countable" orthonormal basis?

2. Since $L^2(\Bbb R^d)$ is a Hilbert space, the above question is a special case of Bounded sequence in a Hilbert space has weakly convergent subsequence (if 1 is true). However, I am curious that is there a more simple proof using that $H$ is explicitly given as $L^2$ in this situation?