= Solution
Define $U:\ell_{1,2}\to\ell_1$ by $(U\alpha)_j=j^2\alpha_j$. This is a linear bijection, with inverse $(U^{-1}\gamma)_j=j^{-2}\gamma_j$, and
$$
\boxed{\|U\alpha\|_1=\|\alpha\|_{1,2}.}
$$
Thus $U$ is an <isometric isomorphism of normed spaces> onto the <absolutely summable sequence space>, which is a <Banach space>. Explicitly, a <Cauchy sequence> $\alpha^{(n)}$ maps to a <Cauchy sequence> $U\alpha^{(n)}\to\gamma\in\ell_1$; setting $\alpha=U^{-1}\gamma$ gives <convergence> in the weighted <norm>. Hence \b[$\ell_{1,2}$ is a <Banach space>], an instance of <weighted summable sequence space isometry>.
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