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  "body": "\n## Summary\n\nA separable Banach space is a [[Banach space|06GZ956657]] that has a countable dense subset.\nMost spaces met in analysis are of this kind, for example $\\ell^p$ and $L^p[0, 1]$ for $1 \\le p \u003c \\infty$, $c_0$ and $C[0, 1]$.\nSeparability allows arguments by sequences and diagonal extraction.\nFor instance, the closed unit ball of the dual of a separable Banach space is metrizable in the weak* topology.\nThe [[Banach–Mazur theorem|QC5MB5WM8F]] says that every separable Banach space is isometric to a closed subspace of $C[0, 1]$.\n\n## Explanations\n\n### L1 human\n\nA space is separable if a countable list of its points comes as close as you like to every point.\nThe rationals play this role in the real line.\nIn the space $C[0, 1]$ of continuous functions with the sup norm, the polynomials with rational coefficients form such a list.\nThe space $\\ell^\\infty$ of bounded sequences is too big: it has uncountably many points at distance $1$ from each other.\n",
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