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 .
0.0.1 Remarks
-
1.
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).
-
2.
With the usual point-wise addition and point-wise multiplication by a scalar, is a vector space over the field .
-
3.
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].
References
-
1
W. Rudin, Functional Analysis
, McGraw-Hill Book Company, 1973.
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 |