smooth functions with compact support

DefinitionLet $U$ be an open set in ${ℝ}^n$. Then the set ofsmooth functions with compact support (in $U$) is the setof functions $f:{ℝ}^n\to ℂ$ which are smooth(i.e., ${\partial }^{\alpha }f:{ℝ}^n\to ℂ$is a continuous function for all multi-indices $\alpha$)and $\mathrm{supp}f$ is compact and contained in $U$.This function space is denoted by $C_0^{\mathrm{\infty }}(U)$.

0.0.1 Remarks

1. A proof that $C_0^{\mathrm{\infty }}(U)$ is non-trivial (that is, it contains other functionsthan the zero function) can be foundhere (http://planetmath.org/Cinfty_0UIsNotEmpty).

2. With the usual point-wise addition and point-wise multiplicationby a scalar, $C_0^{\mathrm{\infty }}(U)$ is a vector space over the field $ℂ$.

3. Suppose $U$ and $V$ are open subsets in ${ℝ}^n$ and $U\subset V$.Then $C_0^{\mathrm{\infty }}(U)$ is a vector subspace of $C_0^{\mathrm{\infty }}(V)$.In particular, $C_0^{\mathrm{\infty }}(U)\subset C_0^{\mathrm{\infty }}(V)$.

It is possible to equip $C_0^{\mathrm{\infty }}(U)$ with a topology, which makes$C_0^{\mathrm{\infty }}(U)$ into a locally convex topological vector space. The idea isto exhaust $U$ with compact sets. Then, for each compact set $K\subset U$,one defines a topology of smooth functions on $U$ withsupport on $K$. The topology for $C_0^{\mathrm{\infty }}(U)$ is the inductivelimit topology of these topologies. See e.g. [1].