1Normed vector spaces
II Linear Analysis
1.4 The double dual
Definition
(Double dual)
.
Let
V
be a normed vector space. Define
V
∗∗
= (
V
∗
)
∗
.
We want to define a map
φ
:
V → V
∗∗
. Again, we can reason about what
we expect this function to do. It takes in a
v ∈ V
, and produces a
φ
(
v
)
∈ V
∗∗
.
Expanding the definition, this gives a
φ
(
v
) :
V
∗
→ F
. Hence this
φ
(
v
) takes in
a g ∈ V
∗
, and returns a φ(v)(g) ∈ F.
This is easy. Since
g ∈ V
∗
, we know that
g
is a function
g
:
V → F
. Given
this function
g
and a
v ∈ V
, it is easy to produce a
φ
(
v
)(
g
)
∈ F
. Just apply
g
on v:
φ(v)(g) = g(v).
Proposition.
Let
φ
:
V → V
∗∗
be defined by
φ
(
v
)(
g
) =
g
(
v
). Then
φ
is a
bounded linear map and kφk
B(V,V
∗
)
≤ 1
Proof. Again, we are taking supremum over non-zero elements. We have
kφk
B(V,V
∗
)
= sup
v∈V
kφ(v)k
V
∗∗
kvk
V
= sup
v∈V
sup
g∈V
∗
|φ(v)(g)|
kvk
V
kgk
V
∗
= sup
v∈V
sup
g∈V
∗
|g(v)|
kvk
V
kgk
V
∗
≤ 1.
In fact, we will later show that kφk
B(V,V
∗
)
= 1.