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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $\\mathcal{S}(\\R^n)$ be the [[Schwartz space|CG9JEATW3Q]] of smooth functions $\\varphi \\colon \\R^n \\to \\C$ such that\n$$p_{\\alpha,\\beta}(\\varphi) = \\sup_{x \\in \\R^n} \\abs{x^\\alpha \\partial^\\beta \\varphi(x)} \u003c \\infty$$\nfor all multi-indices $\\alpha$ and $\\beta$.\nA *tempered distribution* on $\\R^n$ is a linear map $T \\colon \\mathcal{S}(\\R^n) \\to \\C$ for which there are a constant $C \\ge 0$ and an integer $N \\ge 0$ such that\n$$\\abs{T(\\varphi)} \\le C \\sum_{\\abs{\\alpha} \\le N, \\, \\abs{\\beta} \\le N} p_{\\alpha,\\beta}(\\varphi) \\quad \\text{for all } \\varphi \\in \\mathcal{S}(\\R^n).$$\nThe space of tempered distributions is written $\\mathcal{S}'(\\R^n)$.\n\n## Remarks\n\nThe inequality says that $T$ is a [[continuous linear map|03TM135G0N]] for the Fréchet topology of $\\mathcal{S}(\\R^n)$ given by the seminorms $p_{\\alpha,\\beta}$.\nEquivalently, $T(\\varphi_k) \\to 0$ whenever $p_{\\alpha,\\beta}(\\varphi_k) \\to 0$ for all $\\alpha, \\beta$.\n\nExample: the Dirac delta $\\delta(\\varphi) = \\varphi(0)$ is tempered.\nIt is linear, and $\\abs{\\delta(\\varphi)} \\le p_{0,0}(\\varphi)$, so the condition holds with $C = 1$ and $N = 0$.\nMore generally, if $f$ is measurable and $\\int \\abs{f(x)} (1 + \\abs{x})^{-m} \\, dx \u003c \\infty$ for some $m \\ge 0$, then $\\varphi \\mapsto \\int f \\varphi \\, dx$ is tempered.\n\nNon-example: on $\\R$, the function $\\varphi(x) = e^{-\\sqrt{1 + x^2}}$ lies in $\\mathcal{S}(\\R)$, but $\\int e^x \\varphi(x) \\, dx = \\infty$.\nSo the formula $\\varphi \\mapsto \\int e^x \\varphi \\, dx$ does not define a linear map on all of $\\mathcal{S}(\\R)$, and $e^x$ gives no tempered distribution.\nIt still defines a distribution on compactly supported test functions.\n\nDerivatives act by $(\\partial^\\beta T)(\\varphi) = (-1)^{\\abs{\\beta}} T(\\partial^\\beta \\varphi)$, and the Fourier transform acts by $\\hat{T}(\\varphi) = T(\\hat{\\varphi})$.\n",
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