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3.1 Classical Function Spaces and Distributions 97

∞
convergence defines a locally convex topology on C (Ω) (see Treves [305]).
0
∞
With this concept of convergence, C (Ω) is a locally convex topological
0
vector space called D(Ω).
m
m
Similarly, one can define C (Ω) to be the space of functions in C (Ω)
which, together with their derivatives of order ≤ m, have continuous exten-
m
sions to Ω = Ω ∪ ∂Ω.If Ω is bounded and m< ∞, then C (Ω)isaBanach
space with the norm
α

u
= sup |D u(x)|.
C m (Ω)
x∈Ω
|α|≤m
The Spaces C m,α (Ω)
In Section 1.2, we have already introduced H¨older spaces, however, for
completeness, we restate the definition here again. Let Ω be a subset of IR n
and 0 <α ≤ 1. A function u defined on Ω is said to be H¨older continuous
with exponent α in Ω if 0 <α< 1 and if there exists a constant c ≥ 0
such that
α
|u(x) − u(y)|≤ c|x − y|
for all x, y, ∈ Ω. The quantity

|u(x) − u(y)|
[u] α;Ω := sup α
x,y∈Ω |x − y|
x =y
is called the H¨older modulus of u.For α =1,u is called Lipschitz continuous
and [u] 1;Ω is called the Lipschitz modulus. We say that u is locally H¨older
or Lipschitz continuous with exponent α on Ω if it is H¨older or Lipschitz
continuous with exponent α on every compact subset of Ω, respectively. By
m
C m,α (Ω),m ∈ IN 0 , 0 <α ≤ 1, we denote the space of functions in C (Ω)
whose m–th order derivatives are locally H¨older or Lipschitz continuous with
n
exponent α on the open subset Ω ⊂ IR .WeremarkthatH¨older continuity
may be viewed as a fractional differentiability.
m
Further, by C m,α (Ω) we denote the subspace of C (Ω) consisting of
functions which have m–th order H¨older or Lipschitz continuous derivatives
of exponent α in Ω.If Ω is bounded, we define the H¨older or Lipschitz
norm by
β β
|D u(x) − D u(y)|

u
:=
u
+ sup (3.1.1)
C m,α (Ω) C m (Ω) |x − y| α
|β|=m y∈Ω
x =y
The space C m,α (Ω) equipped with the norm || · || is a Banach space.
C m,α (Ω)
Distributions
Let v be a linear functional on D(Ω). Then we denote by v, ϕ the image
of ϕ ∈D(Ω).