= Weighted summable sequence space isometry
{title2=$U(\alpha)_j=w_j\alpha_j$}
For positive weights $w_j$, the <sequence space> with <norm> $\|\alpha\|=\sum_jw_j|\alpha_j|$ is isometrically isomorphic to $\ell_1$: multiplication by $w_j$ is a linear bijection onto the <absolutely summable sequence space>, with inverse multiplication by $w_j^{-1}$. Thus it is a separable <Banach space>. Density can also be seen directly by truncating a weighted summable sequence and then approximating its finitely many coordinates by rational numbers.
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