Schwartz space

<h2 id="definition">Definition</h2>
Let 
 
 
 
 
 N
 
 
 
 {\displaystyle \mathbb {N} }
 
 be the <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> of non-negative <a href="/facts/Integer/PUwwrawx">integers</a>, and for any 
 
 
 
 n
 ∈
 
 N
 
 
 
 {\displaystyle n\in \mathbb {N} }
 
, let 
 
 
 
 
 
 N
 
 
 n
 
 
 :=
 
 
 
 
 
 N
 
 ×
 ⋯
 ×
 
 N
 
 
 ⏟
 
 
 
 n
 
  times
 
 
 
 
 
 {\displaystyle \mathbb {N} ^{n}:=\underbrace {\mathbb {N} \times \dots \times \mathbb {N} } _{n{\text{ times}}}}
 
 be the n-fold <a href="/facts/Cartesian_product/BfqBWzdL">Cartesian product</a>. 
The Schwartz space or space of rapidly decreasing functions on 
 
 
 
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {R} ^{n}}
 
 is the function space
 
 
 
 
 
 S
 
 
 
 (
 
 
 
 R
 
 
 n
 
 
 ,
 
 C
 
 
 )
 
 :=
 
 {
 
 f
 ∈
 
 C
 
 ∞
 
 
 (
 
 
 R
 
 
 n
 
 
 ,
 
 C
 
 )
 ∣
 ∀
 
 α
 
 ,
 
 β
 
 ∈
 
 
 N
 
 
 n
 
 
 ,
 ‖
 f
 
 ‖
 
 
 α
 
 ,
 
 β
 
 
 
 <
 ∞
 
 }
 
 ,
 
 
 {\displaystyle {\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\},}
 
where 
 
 
 
 
 C
 
 ∞
 
 
 (
 
 
 R
 
 
 n
 
 
 ,
 
 C
 
 )
 
 
 {\displaystyle C^{\infty }(\mathbb {R} ^{n},\mathbb {C} )}
 
 is the function space of <a href="/facts/Smooth_function/mffL4ch2">smooth functions</a> from 
 
 
 
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {R} ^{n}}
 
 into 
 
 
 
 
 C
 
 
 
 {\displaystyle \mathbb {C} }
 
, and
 
 
 
 ‖
 f
 
 ‖
 
 
 α
 
 ,
 
 β
 
 
 
 :=
 
 sup
 
 
 x
 
 ∈
 
 
 R
 
 
 n
 
 
 
 
 
 |
 
 
 
 x
 
 
 α
 
 
 (
 
 
 D
 
 
 β
 
 
 f
 )
 (
 
 x
 
 )
 
 |
 
 .
 
 
 {\displaystyle \|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|.}
 
 Here, 
 
 
 
 sup
 
 
 {\displaystyle \sup }
 
 denotes the <a href="/facts/Supremum/oXCKVopz">supremum</a>, and we used <a href="/facts/Multi-index_notation/2wjdPgzS">multi-index notation</a>, i.e. 
 
 
 
 
 
 x
 
 
 α
 
 
 :=
 
 x
 
 1
 
 
 
 α
 
 1
 
 
 
 
 
 x
 
 2
 
 
 
 α
 
 2
 
 
 
 
 …
 
 x
 
 n
 
 
 
 α
 
 n
 
 
 
 
 
 
 {\displaystyle {\boldsymbol {x}}^{\boldsymbol {\alpha }}:=x_{1}^{\alpha _{1}}x_{2}^{\alpha _{2}}\ldots x_{n}^{\alpha _{n}}}
 
 and 
 
 
 
 
 D
 
 β
 
 
 :=
 
 ∂
 
 1
 
 
 
 β
 
 1
 
 
 
 
 
 ∂
 
 2
 
 
 
 β
 
 2
 
 
 
 
 …
 
 ∂
 
 n
 
 
 
 β
 
 n
 
 
 
 
 
 
 {\displaystyle D^{\boldsymbol {\beta }}:=\partial _{1}^{\beta _{1}}\partial _{2}^{\beta _{2}}\ldots \partial _{n}^{\beta _{n}}}
 
.
To put common language to this definition, one could consider a rapidly decreasing function as essentially a function f(x) such that f(x), f ′(x), f ′′(x), ... all exist everywhere on R and go to zero as x→ ±∞ faster than any reciprocal power of x. In particular, 𝒮(Rn, C) is a <a href="/facts/Linear_subspace/2wtKTgHL">subspace</a> of the function space C∞(Rn, C) of smooth functions from Rn into C.

<h2 id="examples-of-functions-in-the-schwartz-space">Examples of functions in the Schwartz space</h2>
<ul><li>If 
 
 
 
 
 α
 
 
 
 {\displaystyle {\boldsymbol {\alpha }}}
 
 is a multi-index, and a is a positive <a href="/facts/Real_number/R02gw5Pb">real number</a>, then

x
 
 
 α
 
 
 
 e
 
 −
 a
 
 |
 
 
 x
 
 
 
 |
 
 
 2
 
 
 
 
 ∈
 
 
 S
 
 
 (
 
 
 R
 
 
 n
 
 
 )
 .
 
 
 {\displaystyle {\boldsymbol {x}}^{\boldsymbol {\alpha }}e^{-a|{\boldsymbol {x}}|^{2}}\in {\mathcal {S}}(\mathbb {R} ^{n}).}
 
</li>
<li>Any smooth function f with <a href="/facts/Compact_support/BTuMGbsb">compact support</a> is in 𝒮(Rn). This is clear since any derivative of f is <a href="/facts/Continuous_function/sKbl02pB">continuous</a> and supported in the support of f, so (
 
 
 
 
 
 x
 
 
 α
 
 
 
 
 D
 
 
 α
 
 
 )
 f
 
 
 {\displaystyle {\boldsymbol {x}}^{\boldsymbol {\alpha }}{\boldsymbol {D}}^{\boldsymbol {\alpha }})f}
 
 has a maximum in Rn by the <a href="/facts/Extreme_value_theorem/eSV6P7eh">extreme value theorem</a>.</li>
<li>Because the Schwartz space is a vector space, any polynomial 
 
 
 
 ϕ
 (
 
 x
 
 )
 
 
 {\displaystyle \phi ({\boldsymbol {x}})}
 
 can be multiplied by a factor 
 
 
 
 
 e
 
 −
 a
 |
 
 x
 
 
 |
 
 2
 
 
 
 
 
 
 {\displaystyle e^{-a\vert {\boldsymbol {x}}\vert ^{2}}}
 
 for 
 
 
 
 a
 >
 0
 
 
 {\displaystyle a>0}
 
 a real constant, to give an element of the Schwartz space. In particular, there is an embedding of polynomials into a Schwartz space.</li></ul>
<h2 id="properties">Properties</h2>
<h3>Analytic properties</h3>
<ul><li>From <a href="/facts/General_Leibniz_rule/zaTBxsRF">Leibniz's rule</a>, it follows that 𝒮(Rn) is also closed under <a href="/facts/Pointwise_product/781oq7Xu">pointwise multiplication</a>:
If f, g ∈ 𝒮(Rn) then the product fg ∈ 𝒮(Rn).</li></ul>
In particular, this implies that 𝒮(Rn) is an R-algebra. More generally, if f ∈ 𝒮(R) and H is a bounded smooth function with bounded derivatives of all orders, then fH ∈ 𝒮(R).

<ul><li>The Fourier transform is a <a href="/facts/Linear_isomorphism/Q1PBKNVV">linear isomorphism</a> F:𝒮(Rn) → 𝒮(Rn).</li>
<li>If f ∈ 𝒮(Rn) then f is <a href="/facts/Lipschitz_continuous/Hw20EPEU">Lipschitz continuous</a> and hence <a href="/facts/Uniformly_continuous/hjvw7tII">uniformly continuous</a> on Rn.</li>
<li>𝒮(Rn) is a <a href="/facts/Distinguished_space/0f3kn6lb">distinguished</a> <a href="/facts/Locally_convex_topological_vector_space/AGrPrayC">locally convex</a> <a href="/facts/Fr%25C3%25A9chet_space/9gXkQu2u">Fréchet</a> <a href="/facts/Schwartz_topological_vector_space/c559KL0o">Schwartz TVS</a> over the <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a>.</li>
<li>Both 𝒮(Rn) and its <a href="/facts/Strong_dual_space/4RI1SYl6">strong dual space</a> are also:</li></ul>
<ol><li><a href="/facts/Complete_topological_vector_space/e6qtS76j">complete</a> <a href="/facts/Hausdorff_space/gbKIrusl">Hausdorff</a> <a href="/facts/Locally_convex_topological_vector_space/AGrPrayC">locally convex</a> spaces,</li>
<li><a href="/facts/Nuclear_space/4FLPWdfp">nuclear</a> <a href="/facts/Montel_space/HLXqXHzs">Montel spaces</a>,</li>
<li><a href="/facts/Ultrabornological_space/DLUE8S0K">ultrabornological spaces</a>,</li>
<li><a href="/facts/Reflexive_space/g7Cjbw0S">reflexive</a> <a href="/facts/Barrelled_space/r2H58TN3">barrelled</a> <a href="/facts/Mackey_space/X2d6rBTI">Mackey spaces</a>.</li></ol>
<h3>Relation of Schwartz spaces with other topological vector spaces</h3>
<ul><li>If 1 ≤ p ≤ ∞, then 𝒮(Rn) ⊂ <a href="/facts/Lp_space/gbad0Rv1">Lp(Rn)</a>.</li>
<li>If 1 ≤ p < ∞, then 𝒮(Rn) is <a href="/facts/Dense_set/nPT9Yxao">dense</a> in Lp(Rn).</li>
<li>The space of all <a href="/facts/Bump_function/ExdBtMHN">bump functions</a>, C∞c(Rn), is included in 𝒮(Rn).</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Bump_function/ExdBtMHN">Bump function</a></li>
<li><a href="/facts/Schwartz%25E2%2580%2593Bruhat_function/T54DNFMp">Schwartz–Bruhat function</a></li>
<li><a href="/facts/Nuclear_space/4FLPWdfp">Nuclear space</a></li></ul>

<h3>Sources</h3>
<ul><li><a href="/facts/Lars_H%25C3%25B6rmander/mxyNnmRA">Hörmander, L.</a> (1990). The Analysis of Linear Partial Differential Operators I, (Distribution theory and Fourier Analysis) (2nd ed.). Berlin: Springer-Verlag. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-52343-X.</li>
<li>Reed, M.; Simon, B. (1980). Methods of Modern Mathematical Physics: Functional Analysis I (Revised and enlarged ed.). San Diego: Academic Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-12-585050-6.</li>
<li>Stein, Elias M.; Shakarchi, Rami (2003). Fourier Analysis: An Introduction (Princeton Lectures in Analysis I). Princeton: Princeton University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-691-11384-X.</li>
<li><a href="/facts/Fran%25C3%25A7ois_Tr%25C3%25A8ves/cpjQbAov">Trèves, François</a> (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-45352-1. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/853623322">853623322</a>.</li></ul>
This article incorporates material from Space of rapidly decreasing functions on <a href="/facts/PlanetMath/wphPCtLf">PlanetMath</a>, which is licensed under the Creative Commons Attribution/Share-Alike License.

Schwartz space open-in-new

Schwartz space