= Solution
Let $D$ be the set of <finitely supported sequences> with rational coordinates. It is countable: it is the union over $N$ of copies of $\mathbb Q^N$. Given $\alpha\in\ell_{1,2}$ and $\varepsilon>0$, choose $N$ with $\sum_{j>N}j^2|\alpha_j|<\varepsilon/2$. Choose rational numbers $q_j$ for $1\leq j\leq N$ so that $\sum_{j=1}^Nj^2|q_j-\alpha_j|<\varepsilon/2$, and put $q_j=0$ thereafter. Then $q\in D$ and $\|q-\alpha\|_{1,2}<\varepsilon$. Thus \b[$D$ is countable and dense, so the space is <separable>].
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