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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]], '''Schwartz space <math>\mathcal{S}</math>''' is the [[function space]] of all [[Function (mathematics)|functions]] whose [[derivative]]s are [[Vanish_at_infinity#Rapidly_decreasing|rapidly decreasing]].  This space has the important property that the [[Fourier transform]] is an [[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]] is an example of a rapidly decreasing function.]]

Schwartz space is named after French mathematician [[Laurent Schwartz]].

==Definition==

Let <math>\mathbb{N}</math> be the [[Set (mathematics)|set]] of non-negative [[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">\mathcal{S} \left(\mathbb{R}^n, \mathbb{C}\right) := \left \{ f \in C^\infty(\mathbb{R}^n, \mathbb{C}) \mid \forall \boldsymbol{\alpha},\boldsymbol{\beta}\in\mathbb{N}^n, \|f\|_{\boldsymbol{\alpha},\boldsymbol{\beta}}< \infty\right \},</math>where <math>C^{\infty}(\mathbb{R}^n, \mathbb{C})</math> is the function space of [[smooth function]]s from <math>\mathbb{R}^n</math> into <math>\mathbb{C}</math>, and<math display="block">\|f\|_{\boldsymbol{\alpha},\boldsymbol{\beta}}:= \sup_{\boldsymbol{x}\in\mathbb{R}^n} \left| \boldsymbol{x}^\boldsymbol{\alpha} (\boldsymbol{D}^\boldsymbol{\beta} f)(\boldsymbol{x}) \right|.</math> Here, <math>\sup</math> denotes the [[supremum]], and we used [[multi-index notation]], i.e. <math>\boldsymbol{x}^\boldsymbol{\alpha}:=x_1^{\alpha_1}x_2^{\alpha_2}\ldots x_n^{\alpha_n}</math> and <math>D^\boldsymbol{\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, {{math|𝒮('''R'''{{sup|''n''}}, '''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 <math>\boldsymbol{\alpha}</math> is a multi-index, and ''a'' is a positive [[real number]], then
*:<math>\boldsymbol{x}^\boldsymbol{\alpha} e^{-a |\boldsymbol{x}|^2} \in \mathcal{S}(\mathbb{R}^n).</math>
* Any smooth function ''f'' with [[compact support]] is in {{math|𝒮('''R'''{{sup|''n''}})}}. This is clear since any derivative of ''f'' is [[Continuous function|continuous]] and supported in the support of ''f'', so (<math>\boldsymbol{x}^\boldsymbol{\alpha}\boldsymbol{D}^\boldsymbol{\alpha})f</math> has a maximum in '''R'''<sup>''n''</sup> by the [[extreme value theorem]].
* Because the Schwartz space is a vector space, any polynomial <math>\phi(\boldsymbol{x})</math> can be multiplied by a factor <math> e^{-a \vert\boldsymbol{x}\vert^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 into 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''}})}}.
In particular, this implies that {{math|𝒮('''R'''{{sup|''n''}})}} is an {{math|'''R'''}}-algebra. More generally, if {{math|''f'' ∈ 𝒮('''R''')}} and {{mvar|H}} is a bounded smooth function with bounded derivatives of all orders, then {{math|''fH'' ∈ 𝒮('''R''')}}.
* The Fourier transform is a [[linear isomorphism]] {{math|F:𝒮('''R'''{{sup|''n''}}) → 𝒮('''R'''{{sup|''n''}})}}.
* If {{math|''f'' ∈ 𝒮('''R'''{{sup|''n''}})}} then {{mvar|f}} is [[Lipschitz continuous]] and hence [[uniformly continuous]] on {{math|'''R'''{{sup|''n''}}}}.
*{{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]]s.
* Both {{math|𝒮('''R'''{{sup|''n''}})}} '''''and''''' its [[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 [[Continuous dual space|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}} - Relevance to this entry? This rather belongs to [[Montel space]] -->
#[[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]]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 |author-link=Lars Hörmander |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}}

{{PlanetMath attribution|id=4444|title=Space of rapidly decreasing functions}}

{{Functional analysis}}

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