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\begin{document}

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\section*{Schwartz space}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{functional_analysis}{}\paragraph*{{Functional analysis}}\label{functional_analysis}

[[!include functional analysis - contents]]

\hypertarget{harmonic_analysis}{}\paragraph*{{Harmonic analysis}}\label{harmonic_analysis}

[[!include harmonic analysis - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In [[functional analysis]],

\begin{enumerate}%
\item \emph{a} [[Schwartz space]] (\hyperlink{Terzioglu69}{Terzioglu 69}, \hyperlink{KrieglMichor97}{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]] $E_V$.


\item \emph{the} Schwartz space of an [[open subset]] of [[Euclidean space]] is the space of [[functions with rapidly decreasing partial derivatives]] (def. \ref{RapidlyDecreasingFunction} below). On this space the operation of [[Fourier transform]] is a linear automorphism (prop. \ref{FourierTransformIsIsomorphismOnSchwartzSpace} below). The [[continuous linear functionals]] on this space are the [[tempered distributions]].



\end{enumerate}
The Schwartz spaces in the second sense are examples of the Schwartz spaces in the first sense.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{RapidlyDecreasingFunction}\hypertarget{RapidlyDecreasingFunction}{}
\textbf{([[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 \emph{[[rapidly decreasing function|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]]:

\begin{displaymath}
\underset{\alpha, \beta \in \mathbb{N}^n}{\forall}
  \left(
  \underset{x \in \mathbb{R}^n}{sup} {\Vert x^\alpha \partial_{\beta} f(x)  \Vert} \lt K_{\alpha, \beta}
  \right)
\end{displaymath}
for some choices of positive constants $K_{\alpha, \beta}$.

\end{defn}
(e.g. \hyperlink{Hoermander90}{H\"o{}rmander 90, def. 7.1.2})

\begin{defn}
\label{SchwartzSpaceOfFunctionsWithRapidlyDecreasingDerivatives}\hypertarget{SchwartzSpaceOfFunctionsWithRapidlyDecreasingDerivatives}{}
\textbf{(Schwartz space of [[functions with rapidly decreasing partial derivatives]])}

For $n \in \mathbb{N}$ the \emph{Schwartz space} $\mathcal{S}(\mathbb{R}^n)$ is the [[topological vector space]] whose

\begin{itemize}%
\item 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. \ref{RapidlyDecreasingFunction});


\item whose [[topological space|topology]] is that induced b the [[semi-norms]] given by $p_{\alpha,\beta}(f) \coloneqq {\Vert x^\alpha \partial_{\beta} (f)(x)  \Vert}$ is called the \emph{Schwartz space} $\mathcal{S}(\mathbb{R}^n)$.



\end{itemize}
\end{defn}
(e.g. \hyperlink{Hoermander90}{H\"o{}rmander 90, def. 7.1.2})

\begin{prop}
\label{SchwartSpaceOfFunctionsIsFrechetSpace}\hypertarget{SchwartSpaceOfFunctionsIsFrechetSpace}{}
\textbf{(the Schwartz space is a [[Fréchet space]])}

The Schwartz space $\mathcal{S}$ of [[functions with rapidly decreasing partial derivatives]] (def. \ref{SchwartzSpaceOfFunctionsWithRapidlyDecreasingDerivatives}) is a [[Fréchet space]].

\end{prop}
(e.g. p. 2 here: \href{https://www.math.ucdavis.edu/~hunter/m218a_09/ch5A.pdf}{pdf})

\begin{defn}
\label{TemperedDistribution}\hypertarget{TemperedDistribution}{}
\textbf{([[tempered distributions]])}

A \emph{[[tempered distribution]]} on $\mathbb{R}^n$ is a [[continuous linear functional]] on the Schwartz space $\mathcal{S}(\mathbb{R}^n)$ (def. \ref{SchwartzSpaceOfFunctionsWithRapidlyDecreasingDerivatives}).

\end{defn}
(e.g. \hyperlink{Hoermander90}{H\"o{}rmander 90, def. 7.1.7})

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{prop}
\label{FourierTransformIsIsomorphismOnSchwartzSpace}\hypertarget{FourierTransformIsIsomorphismOnSchwartzSpace}{}
\textbf{([[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. \ref{RapidlyDecreasingFunction}) and indeed constitutes a [[linear isomorphism]] from the Schwartz space (def. \ref{SchwartzSpaceOfFunctionsWithRapidlyDecreasingDerivatives}) to itself:

\begin{displaymath}
\widehat {(-)}
  \;\colon\;
  \mathcal{S}(\mathbb{R}^n)
    \longrightarrow
  \mathcal{S}(\mathbb{R}^n)
\end{displaymath}
\end{prop}
(e.g. \hyperlink{Hoermander90}{H\"o{}rmander 90, lemma 7.1.3}, \hyperlink{Melrose03}{Melrose 03, theorem 1.3})

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[Schwartz-Bruhat function]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Named by [[Alexander Grothendieck]] after [[Laurent Schwartz]] (according to \hyperlink{Terzioglu69}{Terzioglu 69}).

The general concept of Schwartz spaces appears in

\begin{itemize}%
\item Horvath, 1966, p277


\item T. Terzioglu, \emph{On Schwartz spaces}, Mathematische Annalen September 1969, Volume 182, Issue 3, pp 236--242 (\href{https://link.springer.com/article/10.1007/BF01350326}{web})


\item Jarchow, 1981, 10.4.3, p202


\item [[Andreas Kriegl]], [[Peter Michor]], on p. 585 after Result 52.24 of \emph{[[The Convenient Setting of Global Analysis]]}, 1997



\end{itemize}
Specifically the Schwartz spaces of [[functions with rapidly decreasing partial derivatives]] are discussed for instance in

\begin{itemize}%
\item [[Lars Hörmander]], section 7.1 of \emph{The analysis of linear partial differential operators}, vol. I, Springer 1983, 1990


\item [[Richard Melrose]], chapter 1 of \emph{Introduction to microlocal analysis}, 2003 (\href{http://www-math.mit.edu/~rbm/iml90.pdf}{pdf})



\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Schwartz_space}{Schwartz space}}

\end{itemize}
[[!redirects smooth function with rapidly decreasing derivatives]] [[!redirects smooth functions with rapidly decreasing derivatives]]

[[!redirects function with rapidly decreasing derivatives]] [[!redirects functions with rapidly decreasing derivatives]] [[!redirects rapidly decreasing derivatives]]

category: functional analysis



\end{document}