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smooth functions with compact support (Definition)

Definition Let $ U$ be an open set in $ \mathbb{R}^n$. Then the set of smooth functions with compact support (in $ U$) is the set of functions $ f:\mathbb{R}^n \to \mathbb{C}$ which are smooth (i.e., $ \partial^\alpha f:\mathbb{R}^n\to\mathbb{C}$ is a continuous function for all multi-indices $ \alpha$) and $ \operatorname{supp} f$ is compact and contained in $ U$. This function space is denoted by $ C^\infty_0(U)$.

Remarks

  1. A proof that $ C^\infty_0(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, $ C^\infty_0(U)$ is a vector space over the field $ \mathbb{C}$.
  3. Suppose $ U$ and $ V$ are open subsets in $ \mathbb{R}^n$ and $ U\subset V$. Then $ C^\infty_0(U)$ is a vector subspace of $ C^\infty_0(V)$. In particular, $ C^\infty_0(U)\subset C^\infty_0(V)$.

It is possible to equip $ C^\infty_0 (U)$ with a topology, which makes $ C^\infty_0 (U)$ into a locally convex topological vector space. The idea is to exhaust $ U$ with compact sets. Then, for each compact set $ K\subset U$, one defines a topology of smooth functions on $ U$ with support on $ K$. The topology for $ C_0^\infty(U)$ is the inductive limit topology of these topologies. See e.g. [1].

References

1
W. Rudin, Functional Analysis, McGraw-Hill Book Company, 1973.



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See Also: $C^n$


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$C^\infty_0(U)$ is not empty (Theorem) by matte
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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
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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 MSC26B05 (Real functions :: Functions of several variables :: Continuity and differentiation questions)

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