smooth functions with compact supportSmoothFunctionsWithCompactSupport2003-07-05 10:41:132007-06-02 02:50:25Definition% this is the default PlanetMath preamble. as your knowledge
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{\bf Definition}
Let $U$ be an open set in $\sR^n$. Then the set of
{\bf smooth functions with compact support} (in $U$) is the set
of functions $f:\sR^n \to \sC$ which are smooth
(i.e., $\partial^\alpha f:\sR^n\to\sC$
is a continuous function for all multi-indices $\alpha$)
and $\operatorname{supp} f$ is compact and contained in $U$.
This function space is denoted by $C^\infty_0(U)$.
\subsubsection{Remarks}
\begin{enumerate}
\item A proof that $C^\infty_0(U)$ is non-trivial (that is, it contains other functions
than the zero function) can be found
\PMlinkname{here}{Cinfty_0UIsNotEmpty}.
\item With the usual point-wise addition and point-wise multiplication
by a scalar, $C^\infty_0(U)$ is a vector space over the field $\sC$.
\item Suppose $U$ and $V$ are open subsets in $\sR^n$ and $U\subset V$.
Then $C^\infty_0(U)$ is a vector subspace of $C^\infty_0(V)$.
In particular, $C^\infty_0(U)\subset C^\infty_0(V)$.
\end{enumerate}
It is possible to equip $\scomp(U)$ with a topology, which makes
$\scomp(U)$ into a locally convex topological vector space. The idea is
to exhaust $U$ with compact sets. Then, for each compact set $K\subset U$,
one defines a topology of smooth functions on $U$ with
support on $K$. The topology for $C_0^\infty(U)$ is the inductive
limit topology of these topologies. See e.g. \cite{rudin_fap}.
\begin{thebibliography}{9}
\bibitem{rudin_fap}
W. Rudin, \emph{Functional Analysis},
McGraw-Hill Book Company, 1973.
\end{thebibliography}