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  "body": "\n## Definition\n\nLet $X$ be a [[normed vector space|03J5RAKB30]] over $\\R$ or $\\C$, and let $M$ be a [[linear subspace|00YC1D652K]] of $X$ that is [[closed|0ZCK8DQYN1]].\nThe *quotient normed space* $X/M$ is the [[quotient vector space|0BR3FY5Q8M]] $X/M = \\set{x + M : x \\in X}$ with the norm\n$$\\norm{x + M} = \\inf_{m \\in M} \\norm{x + m}.$$\nThe map $\\pi \\colon X \\to X/M$, $\\pi(x) = x + M$, is the quotient map.\n\n## Remarks\n\nThe number $\\norm{x + M}$ is the distance from $x$ to $M$.\nClosedness of $M$ is what makes this a norm: $\\norm{x + M} = 0$ means that $x$ is in the closure of $M$, hence in $M$.\nThe quotient map satisfies $\\norm{\\pi(x)} \\le \\norm{x}$ and maps the open unit ball of $X$ onto the open unit ball of $X/M$.\nIf $X$ is a Banach space, then so is $X/M$.\nExample: in $\\R^2$ with the Euclidean norm, let $M = \\set{(a, 0) : a \\in \\R}$; then $\\norm{(a, b) + M} = \\inf_{c} \\sqrt{(a + c)^2 + b^2} = \\abs{b}$, which is a norm on $\\R^2/M$.\nExample: in $\\ell^\\infty$ with $M = c_0$, one has $\\norm{x + c_0} = \\limsup_{n} \\abs{x_n}$.\nNon-example: in $c_0$, the subspace $c_{00}$ of finitely supported sequences is dense but not closed; every $x \\in c_0$ has $\\norm{x + c_{00}} = 0$, so the formula gives only a seminorm on $c_0/c_{00}$.\n",
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