smooth functions with compact support
Definition Let be an open set in . Then the set of smooth functions with compact support (in ) is the set of functions which are smooth (i.e., is a continuous function for all multi-indices ) and is compact and contained in . This function space is denoted by .
A proof that is non-trivial (that is, it contains other functions than the zero function) can be found here (http://planetmath.org/Cinfty_0UIsNotEmpty).
With the usual point-wise addition and point-wise multiplication by a scalar, is a vector space over the field .
Suppose and are open subsets in and . Then is a vector subspace of . In particular, .
It is possible to equip with a topology, which makes into a locally convex topological vector space. The idea is to exhaust with compact sets. Then, for each compact set , one defines a topology of smooth functions on with support on . The topology for is the inductive limit topology of these topologies. See e.g. .
- 1 W. Rudin, Functional Analysis, McGraw-Hill Book Company, 1973.
|Title||smooth functions with compact support|
|Date of creation||2013-03-22 13:44:00|
|Last modified on||2013-03-22 13:44:00|
|Last modified by||matte (1858)|