Definition Let $U$ be an open set in $\sR^n$ . Then the set of 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 $C^\infty_0(U)$ .

Remarks

1. A proof that $C^\infty_0(U)$ is non-trivial (that is, it contains other functions than the zero function) can be found here.
2. With the usual point-wise addition and point-wise multiplication by a scalar, $C^\infty_0(U)$ is a vector space over the field $\sC$ .
3. 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, $C^\infty_0(U)\subset C^\infty_0(V)$ .

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. [1].

References

1

W. Rudin, Functional Analysis, McGraw-Hill Book Company, 1973.