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  "body": "\n## Definition\n\nLet $X$ be a [[Banach space|06GZ956657]] over $\\R$ or $\\C$.\nThe space $X$ is *separable* if it is [[separable|0Z2P41SVDP]] as a metric space.\nThat is, there is a countable subset $D \\subseteq X$ that is [[dense|0BSJSX7ME2]] in $X$.\n\n## Remarks\n\nEquivalently, there is a countable set $S \\subseteq X$ whose linear span is dense in $X$.\nIndeed, finite combinations of elements of $S$ with rational (or Gaussian rational) coefficients then form a countable dense set.\nExample: $\\ell^1$ is separable.\nThe finitely supported sequences with rational entries form a countable set $D$.\nFor $x \\in \\ell^1$ and $\\varepsilon \u003e 0$, choose $N$ with $\\sum_{n \u003e N} \\abs{x_n} \u003c \\varepsilon/2$.\nThen choose rational $q_n$ with $\\sum_{n \\le N} \\abs{x_n - q_n} \u003c \\varepsilon/2$.\nThen $(q_1, \\ldots, q_N, 0, \\ldots) \\in D$ is within $\\varepsilon$ of $x$.\nNon-example: $\\ell^\\infty$ is not separable.\nFor each subset $A$ of $\\N$ its indicator sequence $\\mathbf{1}_A$ lies in $\\ell^\\infty$, and $\\norm{\\mathbf{1}_A - \\mathbf{1}_B}_\\infty = 1$ for $A \\ne B$.\nThe open balls of radius $1/2$ around these uncountably many points are disjoint, so no countable set meets them all.\nIf the dual $X^*$ is separable, then $X$ is separable; the converse fails, as $X = \\ell^1$ with $X^* = \\ell^\\infty$ shows.\n",
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