smooth functions with compact support

Definition Let $U$ be an open set in ${ℝ}^n$. Then the set of smooth functions with compact support (in $U$) is the set of 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 functions than the zero function) can be found here (http://planetmath.org/Cinfty_0UIsNotEmpty).

2. With the usual point-wise addition and point-wise multiplication by 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 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].

Title	smooth functions with compact support
Canonical name	SmoothFunctionsWithCompactSupport
Date of creation	2013-03-22 13:44:00
Last modified on	2013-03-22 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