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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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