smooth functions with compact support

Definition Let U be an open set in n. Then the set of smooth functions with compact support (in U) is the set of functions f:n which are smooth (i.e., αf:n is a continuous functionMathworldPlanetmath for all multi-indices α) and suppf is compactPlanetmathPlanetmath and contained in U. This function space is denoted by C0(U).

0.0.1 Remarks

  1. 1.

    A proof that C0(U) is non-trivial (that is, it contains other functions than the zero function) can be found here (

  2. 2.

    With the usual point-wise addition and point-wise multiplication by a scalar, C0(U) is a vector space over the field .

  3. 3.

    Suppose U and V are open subsets in n and UV. Then C0(U) is a vector subspace of C0(V). In particular, C0(U)C0(V).

It is possible to equip C0(U) with a topologyMathworldPlanetmath, which makes C0(U) into a locally convex topological vector space. The idea is to exhaust U with compact sets. Then, for each compact set KU, one defines a topology of smooth functionsMathworldPlanetmath on U with support on K. The topology for C0(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