a Schwartz space (Terzioglu 69, Kriegl-Michor 97, below 52.24) is a locally convex topological vector space $E$ with the property that whenever $U$ is an absolutely convex neighbourhood of $0$ then it contains another, say $V$, such that $U$ maps to a precompact set in the normed vector space $E_V$.
the Schwartz space of an open subset of Euclidean space is the space of functions with rapidly decreasing partial derivatives (def. below). On this space the operation of Fourier transform is a linear automorphism (prop. below). The continuous linear functionals on this space are the tempered distributions.
The Schwartz spaces in the second sense are examples of the Schwartz spaces in the first sense.
(functions with rapidly decreasing partial derivatives)
For $n \in \mathbb{N}$, a smooth function $f \colon \mathbb{R}^n \to \mathbb{R}$ on the Euclidean space $\mathbb{R}^n$ has rapidly decreasing partial derivatives if the absolute value of the product of any partial derivative $\partial_\beta f$ of the function with any polynomial function is a bounded function:
for some choices of positive constants $K_{\alpha, \beta}$.
(e.g. Hörmander 90, def. 7.1.2)
(Schwartz space of functions with rapidly decreasing partial derivatives)
For $n \in \mathbb{N}$ the Schwartz space $\mathcal{S}(\mathbb{R}^n)$ is the topological vector space whose
underlying real vector space is the subspace of the space $C^\infty(\mathbb{R}^n)$ of smooth functions (with pointwise addition and scalar multiplication) on the functions with rapidly decreasing partial derivatives (def. );
whose topology is that induced by the semi-norms given by $p_{\alpha,\beta}(f) \coloneqq {\Vert x^\alpha \partial_{\beta} (f)(x) \Vert}$ is called the Schwartz space $\mathcal{S}(\mathbb{R}^n)$.
(e.g. Hörmander 90, def. 7.1.2)
(the Schwartz space is a Fréchet space)
The Schwartz space $\mathcal{S}$ of functions with rapidly decreasing partial derivatives (def. ) is a Fréchet space.
(e.g. p. 2 here: pdf)
A tempered distribution on $\mathbb{R}^n$ is a continuous linear functional on the Schwartz space $\mathcal{S}(\mathbb{R}^n)$ (def. ).
(e.g. Hörmander 90, def. 7.1.7)
(Fourier transform is linear automorphism of Schwartz space)
For $n \in \mathbb{N}$ the operation of Fourier transform $f \mapsto \hat f$ is well defined on all smooth functions on $\mathbb{R}^n$ with rapidly decreasing derivatives (def. ) and indeed constitutes a linear isomorphism from the Schwartz space (def. ) to itself:
(e.g. Hörmander 90, lemma 7.1.3, Melrose 03, theorem 1.3)
Named by Alexander Grothendieck after Laurent Schwartz (according to Terzioglu 69).
The general concept of Schwartz spaces appears in
Horvath, 1966, p277
T. Terzioglu, On Schwartz spaces, Mathematische Annalen September 1969, Volume 182, Issue 3, pp 236–242 (web)
Jarchow, 1981, 10.4.3, p202
Andreas Kriegl, Peter Michor, on p. 585 after Result 52.24 of The Convenient Setting of Global Analysis, 1997
Specifically the Schwartz spaces of functions with rapidly decreasing partial derivatives are discussed for instance in
Lars Hörmander, section 7.1 of The analysis of linear partial differential operators, vol. I, Springer 1983, 1990
Richard Melrose, chapter 1 of Introduction to microlocal analysis, 2003 (pdf)
See also
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