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smooth functions with compact support
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(Definition)
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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.
- 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].
- 1
- W. Rudin, Functional Analysis, McGraw-Hill Book Company, 1973.
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"smooth functions with compact support" is owned by matte.
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(view preamble)
See Also: 
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Cross-references: inductive limit, support, smooth functions, compact sets, locally convex topological vector space, topology, vector subspace, field, vector space, scalar, multiplication, addition, contains, function space, contained, compact, multi-indices, continuous function, smooth, functions, open set
There are 7 references to this entry.
This is version 7 of smooth functions with compact support, born on 2003-07-05, modified 2007-06-02.
Object id is 4423, canonical name is SmoothFunctionsWithCompactSupport.
Accessed 6325 times total.
Classification:
| AMS MSC: | 26B05 (Real functions :: Functions of several variables :: Continuity and differentiation questions) |
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Pending Errata and Addenda
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