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  "body": "\n## Definition\n\nLet $\\Omega$ be a set and let $X$ be a vector space of functions $\\Omega \\to \\C$.\nLet $\\varphi \\colon \\Omega \\to \\Omega$ be a map such that $f \\circ \\varphi \\in X$ for every $f \\in X$.\nThe *composition operator* with symbol $\\varphi$ is the [[linear map|0NRYHJPQG2]]\n$$C_\\varphi \\colon X \\to X, \\qquad C_\\varphi f = f \\circ \\varphi,$$\nwhere $\\circ$ is [[function composition|07HF9YCWPP]].\n\n## Remarks\n\nLinearity is automatic: $(af + bg) \\circ \\varphi = a (f \\circ \\varphi) + b (g \\circ \\varphi)$.\n\nExample: let $X$ be the [[Hardy space|PNNNFJVF6W]] $H^2$ of the unit disc $\\mathbb{D}$ and $\\varphi(z) = z/2$.\nFor $f(z) = \\sum_{n \\ge 0} a_n z^n$ in $H^2$, $f \\circ \\varphi = \\sum_{n \\ge 0} 2^{-n} a_n z^n$, whose coefficients are again square-summable; so $C_\\varphi$ is defined, and it is the diagonal operator with entries $2^{-n}$ on the basis $z^n$.\n\nNon-example: $\\varphi(z) = 2z$ does not map $\\mathbb{D}$ into $\\mathbb{D}$.\nFor $f(z) = \\sum_{n \\ge 0} z^n / (n + 1)$, which lies in $H^2$, the function $f(2z)$ is not defined on all of $\\mathbb{D}$, since the series diverges for $\\abs{z} \u003e 1/2$.\n\nWhen $X$ is a normed space one usually asks in addition that $C_\\varphi$ be bounded.\n",
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