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96 3. Representation Formulae

α!= α 1 ! ··· α n !
and
β β! β 1 β n

= = ... .
α (β − α)!α! α 1 α n
We use the usual compact notation for the partial derivatives: If
n
α
α =(α 1 , ··· ,α n ) ∈ IN , we denote by D u the partial derivatives
0
∂ |α| u
α
D u =
∂x ,∂x α 2 α n
α 1
1 2 ...∂x n
α
of order |α|. In particular, if |α| = 0, then D u = u.
Functions with Compact Support
n
Let u be a function defined on an open subset Ω ⊂ IR .The support of
n
u, denoted by supp u, is the closure in IR of the set
{x ∈ Ω : u(x) =0}.
In other words, the support of u is the smallest closed subset of Ω outside of
which u vanishes. We say that u has a compact support in Ω if its support
is a compact (i.e. closed and bounded) subset of Ω. By the notation K Ω,
we mean not only that K ⊂ Ω but also that K is a compact subset of Ω.
Thus, if u has a compact support in Ω, we may write supp u Ω.

m
The Spaces C (Ω)and C 0 ∞ (Ω)
m
We denote by C (Ω),m ∈ IN 0 , the space of all real– (or complex–)valued
n
functions defined in an open subset Ω ⊂ IR having continuous derivatives of
0
order ≤ m.Thus, for m =0,C (Ω) is the space of all continuous functions
in Ω which will be simply denoted by C(Ω). We set
( m
∞
C (Ω)= C (Ω),
m∈IN 0
the space of functions defined in Ω having derivatives of all orders, i.e., the
space of functions which are infinitely differentiable.
∞
We define C (Ω) to be the space of all infinitely differentiable functions
0
which, together with all of their derivatives, have compact support in Ω.We
m
m
denote by C (Ω) the space of functions u ∈ C (Ω) with supp u Ω.The
0
m
∞
spaces C (Ω) as well as C (Ω) are linear function spaces.
0
0
On the linear function space C (Ω) one can introduce the notion of
∞
0
∞ ∞
convergence ϕ j → ϕ in C (Ω) if to the sequence of functions {ϕ j } j=1 there
0
exists a common compact subset K Ω with supp ϕ j ⊂ K for all j ∈
α
α
IN and D ϕ j → D ϕ uniformly for every multi–index α. This notation of