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 $f:{\mathbb{R}}^{n}\to \u2102$ which are smooth (i.e., ${\partial}^{\alpha}f:{\mathbb{R}}^{n}\to \u2102$ is a continuous function^{} for all multiindices $\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 nontrivial (that is, it contains other functions than the zero function) can be found here (http://planetmath.org/Cinfty_0UIsNotEmpty).

2.
With the usual pointwise addition and pointwise multiplication by a scalar, ${C}_{0}^{\mathrm{\infty}}(U)$ is a vector space over the field $\u2102$.

3.
Suppose $U$ and $V$ are open subsets in ${\mathbb{R}}^{n}$ 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 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 ${C}_{0}^{\mathrm{\infty}}(U)$ is the inductive limit topology of these topologies. See e.g. [1].
References
 1 W. Rudin, Functional Analysis^{}, McGrawHill Book Company, 1973.
Title  smooth functions with compact support 

Canonical name  SmoothFunctionsWithCompactSupport 
Date of creation  20130322 13:44:00 
Last modified on  20130322 13:44:00 
Owner  matte (1858) 
Last modified by  matte (1858) 
Numerical id  10 
Author  matte (1858) 
Entry type  Definition 
Classification  msc 26B05 
Related topic  Cn 