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g
1
−
1
To this end, let
g
0
∈
H
\
N
and let
g
1
:=
P
N
g
0
=
:=
g
0
−
×
g
0
. Then
g
(g
0
−
g
1
)
satisfies (
A.12
).
Next, every
v
∈
H
can be written as
v
=
λg
+
w
with
λ
∈ R
and
w
∈
N
: put, for
example,
u
∗
(v)
u
∗
(g)
,w
λ
=
=
v
−
λg .
u
∗
(v)/u
∗
(g)
. Hence
Then, we get 0
=
(g, w)
=
(g, v
−
λg)
,i.e.
(g, v)
=
λ
=
u
∗
(g)g
satisfies
∀
u
:=
u
∗
(v)
u
∗
(λg
λu
∗
(g)
u
∗
(w)
λu
∗
(g)
∈
H
:
=
+
=
+
=
v
w)
(g, v)u
∗
(g)
=
=
(u, v) ,
i.e. (
A.10
) and, by (
A.1
),
u
∗
(v)
|
|
|
|
(u, v)
u
∗
H
∗
=
=
≤
sup
v
sup
v
u
v
v
∈
H
∈
H
and
u
∗
(u)
u
∗
(v)
=
|
|
|
|
(u, u)
u
∗
H
∗
.
=
≤
=
u
sup
v
u
u
v
∈
H
H
∗
is isomorphic to
Remark A.2.2
The preceding result shows that
H
, and the map
u
∗
→
H
∗
u
is, by (
A.11
), an isometry. One therefore
often, but not always
identifies
H
with
.
In the parabolic setting, we often have the following. Let
V
⊂
H
be a subspace
which is dense in
H
and equipped with a norm
·
V
such that the canonical em-
bedding
i
:
V
→
H
is continuous:
∀
∈
V
:
≤
V
.
v
v
C
v
(A.13)
H
∗
. Then one can embed
V
∗
as follows: given
u
We identify
H
and
H
into
∈
H
,
the map
u
u
:
H
∗
and, due to
|
u
∗
(v)
|=|
(u, v)
|≤
u
v
≤
C
u
v
V
V
v
→
(u, v)
is in
∀
v
∈
V
,
V
∗
. We denote by
T
u
u
also in
:
u
→
the mapping with
∀
u
∈
H
∀
v
∈
V
:
(T u)(v)
=
(u, v) .
:
H
→
V
∗
Then
T
satisfies:
|
(T u)(w)
|
|
|
w
V
≤
(u,v)
(i)
Tu
V
∗
=
sup
w
∈
V
=
sup
w
∈
V
C
u
,
w
V
(ii)
T
injective,
(iii)
T(
H
)
is dense in
V
∗
.
V
∗
With
T
we can embed
H
into
and get the triple
→
H
=
H
∗
→
V
∗
,
V
(A.14)
where the canonical injections are continuous and dense, and where
H
is called a
V
∗
'pivot space'. Note that, in (
A.14
), one
cannot
identify
V
and
any more.
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