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= n 0 C n (G) . The support of u is denoted by supp u , and we
define C 0 (G) , C 0
and set C (G)
C n (G) , C (G) with compact
(G) consisting of all functions u
support supp u
G .
We denote by L p (G) ,1
p
≤∞
the usual space which consists of all Lebesgue
with finite L p -norm,
measurable functions u
:
G
→ R
( G |
p d x) 1 /p
u(x)
|
if 1
p<
,
u
L p (G) :=
ess sup G |
u(x)
|
if p
=∞
,
where ess sup means the essential supremum disregarding values on nullsets. The
case p =
2 is of particular interest. The space L 2 (G) is a Hilbert space with respect
to the inner product (u, v)
G u(x)v(x) d x .
=
H
·
·
H :=
Let
be a Hilbert space with the inner product (
,
) H
and norm
u
(u, u) 1 / 2
H
H
. We denote by
the dual space of
H
which consists of all bounded
linear functionals u : H → R
H
on
H
.
can be identified with
H
by the Riesz
representation theorem.
Theorem 2.1.1 (Riesz representation theorem) For each u H
there exists a
unique element u H
such that
u ,v H , H = (u, v) H
v H .
The mapping u u is a linear isomorphism of
H
onto
H
.
The theory of parabolic partial differential equations requires the introduction
of Hilbert space-valued L p -functions. As above, let
H
be a Hilbert space with the
· H
:=
≤∞
norm
. Denote by J the interval J
( 0 ,T) with T> 0, and let 1
p
.
The space L p (J
; H
) is defined by
L p (J
; H
)
:= {
u
:
J
H
measurable
:
u
L p (J ; H ) <
∞}
,
with the norm
( J u(t)
p
H
d t) 1 /p
if 1
p< ,
u L p (J ; H ) :=
ess sup J u(t) H
if p =∞ .
∈ N 0 let C n (J
Furthermore, for n
; H
) be the space of
H
-valued functions that are
of the class C n
with respect to t .
2.2 Partial Differential Equations
For k ∈ N
we let
D k u(x) := { D n u(x) :|
n
|= k }
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