Download PDF

smooth functions with compact support∗
matte†
2013-03-21 16:12:59
Definition Let U be an open set in Rn . Then the set of smooth functions
with compact support (in U ) is the set of functions f : Rn → C which are
smooth (i.e., ∂ α f : Rn → C is a continuous function for all multi-indices α)
and supp f is compact and contained in U . This function space is denoted by
C0∞ (U ).
0.0.1
Remarks
1. A proof that C0∞ (U ) is non-trivial (that is, it contains other functions
than the zero function) can be found here.
2. With the usual point-wise addition and point-wise multiplication by a
scalar, C0∞ (U ) is a vector space over the field C.
3. Suppose U and V are open subsets in Rn and U ⊂ V . 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 topology, 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 K ⊂ U , one defines a topology of smooth
functions on U with support on K. The topology for C0∞ (U ) is the inductive
limit topology of these topologies. See e.g. [?].
References
[1] W. Rudin, Functional Analysis, McGraw-Hill Book Company, 1973.
∗ hSmoothFunctionsWithCompactSupporti
created: h2013-03-21i by: hmattei version:
h34423i Privacy setting: h1i hDefinitioni h26B05i
† This text is available under the Creative Commons Attribution/Share-Alike License 3.0.
You can reuse this document or portions thereof only if you do so under terms that are
compatible with the CC-BY-SA license.
1