smooth functions with compact support

Definition Let $U$ be an open set in $\mathbb{R}^{n}$. Then the set of smooth functions with compact support (in $U$) is the set of functions $f:\mathbb{R}^{n}\to\mathbb{C}$ which are smooth (i.e., $\partial^{\alpha}f:\mathbb{R}^{n}\to\mathbb{C}$ 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 $C^{\infty}_{0}(U)$.

0.0.1 Remarks

1. 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 (http://planetmath.org/Cinfty_0UIsNotEmpty).

2. 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 $\mathbb{C}$.

3. 3.

Suppose $U$ and $V$ are open subsets in $\mathbb{R}^{n}$ 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 $C^{\infty}_{0}(U)$ with a topology  , which makes $C^{\infty}_{0}(U)$ 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. .

References

• 1
Title smooth functions with compact support SmoothFunctionsWithCompactSupport 2013-03-22 13:44:00 2013-03-22 13:44:00 matte (1858) matte (1858) 10 matte (1858) Definition msc 26B05 Cn