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  "body": "\n## Definition\n\nLet $X$ be a [[normed vector space|03J5RAKB30]] over $\\R$ or $\\C$.\nA *completion* of $X$ is a pair $(\\widehat{X}, \\iota)$ such that:\n\n1. $\\widehat{X}$ is a [[Banach space|06GZ956657]] over the same field.\n2. $\\iota \\colon X \\to \\widehat{X}$ is a [[linear isometry|07SJXTWVQW]], that is, $\\iota$ is linear and $\\norm{\\iota(x)} = \\norm{x}$ for all $x \\in X$.\n3. The image $\\iota(X)$ is [[dense|0BSJSX7ME2]] in $\\widehat{X}$.\n\n## Remarks\n\nEvery normed space has a completion: take the Cauchy sequences in $X$ modulo the sequences that tend to $0$, with $\\norm{[(x_n)]} = \\lim_n \\norm{x_n}$, and $\\iota(x)$ the constant sequence.\nAny two completions $(\\widehat{X}, \\iota)$ and $(\\widetilde{X}, \\kappa)$ are related by a unique isometric isomorphism $\\phi$ with $\\phi \\circ \\iota = \\kappa$.\nExample: let $P$ be the space of real polynomial functions on $[0, 1]$ with the sup norm, and let $\\iota$ be the inclusion of $P$ in $C[0, 1]$.\nClauses 1 and 2 hold, and clause 3 holds by the [[Weierstrass approximation theorem|0JQE130DHW]], so $C[0, 1]$ is the completion of $P$.\nNon-example: the inclusion of the finitely supported sequences $c_{00}$, with the sup norm, into $\\ell^\\infty$ satisfies clauses 1 and 2 but not 3.\nThe reason is that the closure of $c_{00}$ in $\\ell^\\infty$ is the space $c_0$ of null sequences.\nThat closure, $c_0$, is the completion of $c_{00}$.\nAs a metric space, $\\widehat{X}$ is the [[completion of the metric space|06RW4RKWH7]] $X$, with the vector operations extended by continuity.\n",
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