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  "body": "\n## Summary\n\nThe Schwartz space $\\mathcal{S}(\\R^n)$ is the space of smooth functions on $\\R^n$ that, together with all their derivatives, decay faster than any power of $1/\\abs{x}$.\nThe [[Fourier transform|DPVM34ADZV]] maps it bijectively onto itself, which makes it the natural home of Fourier analysis on $\\R^n$.\nIts dual space is the space of [[tempered distributions|AES2T5JPFP]].\nSee the [[definition|GE020MY43M]].\n\n## Explanations\n\n### L1 human\n\nA Schwartz function is smooth and dies off very fast at infinity, and so do all its derivatives.\nThe bell curve $e^{-x^2}$ is the basic example: it decays faster than $1/x^{100}$ or any other power.\nThe function $1/(1 + x^2)$ is smooth but decays only like $1/x^2$, so it is not a Schwartz function.\nThe point of the space is that differentiation, multiplication by polynomials and the Fourier transform never lead out of it.\n",
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