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  "body": "\n## Definition\n\nThe *Schwartz space* $\\mathcal{S}(\\R^n)$ is the set of [[smooth functions|0V33BNKZ9S]] $f \\colon \\R^n \\to \\C$ such that\n$$p_{\\alpha, \\beta}(f) = \\sup_{x \\in \\R^n} \\abs{x^\\alpha \\, \\partial^\\beta f(x)} \u003c \\infty$$\nfor all [[multi-indices|0PJXBR6QA3]] $\\alpha, \\beta \\in \\N^n$.\nIts elements are called *Schwartz functions* or *rapidly decreasing functions*.\nWith the family of seminorms $p_{\\alpha, \\beta}$, the space $\\mathcal{S}(\\R^n)$ is a [[Fréchet space|3GAA614673]].\n\n## Remarks\n\nExample: the Gaussian $f(x) = e^{-\\abs{x}^2}$ is a Schwartz function.\nEach derivative $\\partial^\\beta f$ is a polynomial times $e^{-\\abs{x}^2}$, and a polynomial times $e^{-\\abs{x}^2}$ is bounded on $\\R^n$.\nEvery [[test function|HS65DW0HXH]] on $\\R^n$ is a Schwartz function.\nNon-example: $f(x) = 1/(1 + x^2)$ on $\\R$ is smooth, but $p_{2, 0}(f) = \\sup_x x^2/(1 + x^2) = 1$ while $p_{3, 0}(f) = \\sup_x \\abs{x}^3/(1 + x^2) = \\infty$.\nThe space is closed under differentiation, under multiplication by polynomials and under the [[Fourier transform|DPVM34ADZV]], which is a bijection of $\\mathcal{S}(\\R^n)$ onto itself.\nEquivalently, $f$ is a Schwartz function if and only if $(1 + \\abs{x})^N \\abs{\\partial^\\beta f(x)}$ is bounded for every $N \\in \\N$ and every $\\beta$.\n\n## Explanations\n\n### L1 human\n\nA Schwartz function is a smooth function that, with all its derivatives, decays faster than every inverse power of $\\abs{x}$.\n",
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