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{{Short description|Function space of all functions whose derivatives are rapidly decreasing}}
{{for multi|the Schwartz space of a semisimple Lie group|Harish-Chandra's Schwartz space|the Schwartz space of a locally compact abelian group|Schwartz–Bruhat function}}
In [[Mathematics|mathematics]], '''Schwartz space <math>\mathcal{S}</math>''' is the [[Function space|function space]] of all [[Function (mathematics)|functions]] whose [[Derivative|derivative]]s are rapidly decreasing.  This space has the important property that the [[Fourier transform]] is an [[Automorphism|automorphism]] on this space. This property enables one, by duality, to define the Fourier transform for elements in the dual space <math>\mathcal{S}^*</math> of <math>\mathcal{S}</math>, that is, for tempered distributions.  A function in the Schwartz space is sometimes called a '''Schwartz function'''.
[[File:Gaussian 2D.png|right|thumb|250px|A two-dimensional [[Gaussian function|Gaussian function]] is an example of a rapidly decreasing function.]]

Schwartz space is named after French mathematician [[Biography:Laurent Schwartz|Laurent Schwartz]].

==Definition==

Let <math>\mathbb{N}</math> be the [[Set (mathematics)|set]] of non-negative [[Integer|integer]]s, and for any <math>n \in \mathbb{N}</math>, let <math>\mathbb{N}^n := \underbrace{\mathbb{N} \times \dots \times \mathbb{N}}_{n \text{ times}}</math> be the ''n''-fold [[Cartesian product]]. The ''Schwartz space'' or '''space of rapidly decreasing functions on''' <math>\mathbb{R}^n</math> is the function space<math display="block">S \left(\mathbb{R}^n, \mathbb{C}\right) := \left \{ f \in C^\infty(\mathbb{R}^n, \mathbb{C}) \mid \forall \alpha, \beta \in\mathbb{N}^n, \|f\|_{\alpha,\beta} < \infty\right \},</math>where <math>C^{\infty}(\mathbb{R}^n, \mathbb{C})</math> is the function space of smooth functions from <math>\mathbb{R}^n</math> into <math>\mathbb{C}</math>, and<math display="block">\|f\|_{\alpha,\beta}:= \sup_{x\in\mathbb{R}^n} \left| x^\alpha (D^{\beta} f)(x) \right|.</math> Here, <math>\sup</math> denotes the supremum, and we used [[Multi-index notation|multi-index notation]], i.e. <math>x^\alpha:=x_1^{\alpha_1}x_2^{\alpha_2}\ldots x_n^{\alpha_n}</math> and <math>D^\beta:=\partial_1^{\beta_1}\partial_2^{\beta_2}\ldots \partial_n^{\beta_n}</math>.

To put common language to this definition, one could consider a rapidly decreasing function as essentially a function {{math|''f''(''x'')}} such that {{math|''f''(''x'')}}, {{math|''f''{{thin space}}′(''x'')}}, {{math|''f''{{thin space}}′′(''x'')}}, ... all exist everywhere on {{math|'''R'''}} and go to zero as {{mvar|x}}{{math|→ ±∞}} faster than any reciprocal power of {{mvar|x}}. In particular, {{mvar|S}}({{math|'''R'''}}{{sup|{{mvar|n}}}}, {{math|'''C'''}}) is a [[Linear subspace|subspace]] of the function space {{mvar|C}}{{sup|{{math|∞}}}}({{math|'''R'''}}{{sup|{{mvar|n}}}}, {{math|'''C'''}}) of smooth functions from {{math|'''R'''{{sup|{{mvar|n}}}}}} into {{math|'''C'''}}.

==Examples of functions in the Schwartz space==
* If α is a multi-index, and ''a'' is a positive [[Real number|real number]], then
*:<math>x^\alpha e^{-a |x|^2} \in \mathcal{S}(\mathbf{R}^n).</math>
* Any smooth function ''f'' with compact support is in ''S''('''R'''<sup>''n''</sup>). This is clear since any derivative of ''f'' is [[Continuous function|continuous]] and supported in the support of ''f'', so (''x''<sup>''α''</sup>''D''<sup>β</sup>) ''f'' has a maximum in '''R'''<sup>''n''</sup> by the [[Extreme value theorem|extreme value theorem]].
* Because the Schwartz space is a vector space, any polynomial <math>\phi(x^\alpha)</math> can be multiplied by a factor <math> e^{-ax^2}</math> for <math>a > 0</math> a real constant, to give an element of the Schwartz space. In particular, there is an embedding of polynomials inside a Schwartz space.

== Properties ==

=== Analytic properties ===

* From [[General Leibniz rule|Leibniz's rule]], it follows that {{math|𝒮('''R'''{{sup|''n''}})}} is also closed under [[Pointwise product|pointwise multiplication]]:
*: If {{math|''f'', ''g'' ∈ 𝒮('''R'''{{sup|''n''}})}} then the product {{math|''fg'' ∈ 𝒮('''R'''{{sup|''n''}})}}.
* The Fourier transform is a linear isomorphism {{math|F:𝒮('''R'''{{sup|''n''}}) → 𝒮('''R'''{{sup|''n''}})}}.
* If {{math|''f'' ∈ 𝒮('''R''')}} then {{mvar|f}} is uniformly continuous on {{math|'''R'''}}.
*{{math|𝒮('''R'''{{sup|''n''}})}} is a [[Distinguished space|distinguished]] [[Locally convex topological vector space|locally convex]] [[Fréchet space|Fréchet]] [[Schwartz topological vector space|Schwartz TVS]] over the [[Complex number|complex number]]s.
* Both {{math|𝒮('''R'''{{sup|''n''}})}} '''''and''''' its [[Strong dual space|strong dual space]] are also:
#[[Complete topological vector space|complete]] [[Hausdorff space|Hausdorff]] [[Locally convex topological vector space|locally convex]] spaces,
#[[Nuclear space|nuclear]] [[Montel space]]s,
::It is known that in the dual space of any Montel space, a sequence converges in the [[Strong dual space|strong dual topology]] if and only if it converges in the weak* topology,{{sfn | Trèves | 2006 | pp=351–359}}
#[[Ultrabornological space]]s,
#[[Reflexive space|reflexive]] [[Barrelled space|barrelled]] [[Mackey space]]s.

=== Relation of Schwartz spaces with other topological vector spaces ===

*If {{math|1 ≤ ''p'' ≤ ∞}}, then {{math|𝒮('''R'''<sup>''n''</sup>) ⊂ [[Lp space|L<sup>''p''</sup>('''R'''<sup>''n''</sup>)]]}}.
*If {{math|1 ≤ ''p'' < ∞}}, then {{math|𝒮('''R'''<sup>''n''</sup>)}} is [[Dense set|dense]] in {{math|L<sup>''p''</sup>('''R'''<sup>''n''</sup>)}}.
*The space of all [[Bump function|bump function]]s, {{math|C{{su|b=c|p=∞}}('''R'''<sup>''n''</sup>)}}, is included in {{math|𝒮('''R'''<sup>''n''</sup>)}}.

==See also==
* [[Bump function]]
* [[Schwartz–Bruhat function]]
* [[Nuclear space]]

== References ==
<references />

===Sources===
*{{cite book |first=L. |last=Hörmander |title=The Analysis of Linear Partial Differential Operators I, (Distribution theory and Fourier Analysis) |edition=2nd |location=Berlin |publisher=Springer-Verlag |year=1990 |isbn=3-540-52343-X }}
*{{cite book |first=M. |last=Reed |first2=B. |last2=Simon |title=Methods of Modern Mathematical Physics: Functional Analysis I |edition=Revised and enlarged |publisher=Academic Press |location=San Diego |year=1980 |isbn=0-12-585050-6 }}
*{{cite book |first=Elias M. |last=Stein |first2=Rami |last2=Shakarchi |title=Fourier Analysis: An Introduction (Princeton Lectures in Analysis I) |location=Princeton |publisher=Princeton University Press |year=2003 |isbn=0-691-11384-X }}
* {{Trèves François Topological vector spaces, distributions and kernels}}


{{Functional analysis}}

[[Category:Topological vector spaces]]
[[Category:Smooth functions]]
[[Category:Fourier analysis]]
[[Category:Function spaces]]

{{Sourceattribution|Schwartz space}}