Find the weak sequential closure of a set in $L^2(-\pi,\pi)$ $A=\{f_{m,n}(t)|0\le m<n\}$ where $f_{m,n}(t)=e^{imt}+me^{int}$. I should find the weak sequential closure of $A\subset L^2(-\pi,\pi)$. I know what I'm supposed to do. Take a sequence in $A$ and find its weak limit. I also know what is weak convergence and I know that $(L^2)^{*}=L^2$, but I don't know what to do with this:
$\displaystyle\lim\limits_{k\to\infty}\int\limits_{-\pi}^{\pi}(e^{im_kt}+m_ke^{in_kt})f(t)dt=\int\limits_{-\pi}^{\pi}?f(t)dt$ for every $f\in L^2(-\pi,\pi)$.
Thank you very much for any help.
 A: Let $H$ be a separable Hilbert space with orthonormal basis $E=\{e_n\}_{n\in\mathbb{N}}$. Clearly, $(e_n)_{n\in\mathbb{N}}$ weakly converges to $0$. Hence for a fixed $m\in\mathbb{N}$ the sequence $(e_m+me_n)_{n\in\mathbb{N}}$ weakly converges to $e_m$. Hence $E$ is in the weak sequential closure of $S=\{e_m+me_n:m,n\in\mathbb{N}\}$. We denote this closure as $\overline{S}$. Clearly, $S\subset\overline{S}$ and as we showed $E\subset \overline{S}$, so $S\cup E\subset \overline{S}$. Assume $z\in\overline{S}$, then $z$ is weak limit of the sequence $(e_{m_k}+m_k e_{n_k})_{k\in\mathbb{N}}$. Note that $\Vert e_{m_k}+m_k e_{n_k}\Vert\geq m_k-1$. Since weakly convergent sequence are bounded we get that $(m_k)_{k\in\mathbb{N}}$ is bounded, so it has to be stationary (with limit, say m_0) as any subsequence is non-decreasing. If $(n_k)_{k\in\mathbb{N}}$ is also stationary with limit $n_0$, then $(e_{m_k}+m_k e_{n_k})_{k\in\mathbb{N}}$ weakly converges to $e_{m_0}+m_0 e_{n_0}$, so $z=e_{m_0}+m_0 e_{n_0}\in S\subset S\cup E$. If $(n_k)_{k\in\mathbb{N}}$ is not stationary, then $(e_{n_k})_{k\in\mathbb{N}}$ weakly converges to $0$ and we get that $(e_{m_k}+m_k e_{n_k})_{k\in\mathbb{N}}$ weakly converges to $e_{m_0}+m_0\cdot 0$. So $z=e_{m_0}\in E\subset S\cup E$. In both cases we get $z\in S\cup E$. As $z\in \overline{S}$ is arbitrary $\overline{S}\subset S\cup E$. But, as we roved earlier $S\cup E\subset \overline{S}$. Thus $\overline{S}=S\cup E$.
Now you need to apply this "theorem" to $H=L_2(-\pi,\pi)$ with $E=\{e^{int}\}_{n\in\mathbb{N}}$.
