1Normed vector spaces
II Linear Analysis
1.5 Isomorphism
So far, we have discussed a lot about bounded linear maps, which are “morphisms”
between normed vector spaces. It is thus natural to come up with the notion of
isomorphism.
Definition
(Isomorphism)
.
Let
X, Y
be normed vector spaces. Then
T
:
X → Y
is an isomorphism if it is a bounded linear map with a bounded linear inverse
(i.e. it is a homeomorphism).
We say X and Y are isomorphic if there is an isomorphism T : X → Y .
We say that
T
:
X → Y
is an isometric isomorphism if
T
is an isomorphism
and kT xk
Y
= kxk
X
for all x ∈ X.
X
and
Y
are isometrically isomorphic if there is an isometric isomorphism
between them.
Example.
Consider a finite-dimensional space
F
n
with the standard basis
{e
1
, ··· , e
n
}. For any v =
P
v
i
e
i
, the norm is defined by
kvk =
X
v
2
i
1/2
.
Then any
g ∈ V
∗
is determined by
g
(
e
i
) for
i
= 1
, ··· , n
. We want to show that
there are no restrictions on what
g
(
e
i
) can be, i.e. whatever values I assign to
them, g will still be bounded. We have
kgk
V
∗
= sup
v∈V
|g(v)|
kvk
≤ sup
v∈V
P
|v
i
||g(e
i
)|
(
P
|v
i
|
2
)
1/2
≤ C sup
v∈V
(
P
|v
i
|
2
)
1
2
(
P
|v
i
|
2
)
1
2
sup
i
|g(e
i
)|
= C sup
i
|g(e
i
)|
for some
C
, where the second-to-last line is due to the Cauchy-Schwarz inequality.
The supremum is finite since F
n
is finite dimensional.
Since
g
is uniquely determined by a list of values (
g
(
e
1
)
, g
(
e
2
)
, ··· , g
(
e
n
)),
it has dimension
n
. Therefore,
V
∗
is isomorphic to
F
n
. By the same lines of
argument, V
∗∗
is isomorphic to F
n
.
In fact, we can show that
φ
:
V → V
∗∗
by
φ
(
v
)(
g
) =
g
(
v
) is an isometric
isomorphism (this is not true for general normed vector spaces. Just pick
V
to
be incomplete, then V and V
∗∗
cannot be isomorphic since V
∗∗
is complete).
Example. Consider `
p
for p ∈ [1, ∞). What is `
∗
p
?
Suppose q is the conjugate exponent of p, i.e.
1
q
+
1
p
= 1.
(if p = 1, define q = ∞) It is easy to see that `
q
⊆ `
∗
p
by the following:
Suppose (
x
1
, x
2
, ···
)
∈ `
p
, and (
y
1
, y
2
, ···
)
∈ `
q
. Define
y
(
x
) =
P
∞
i=1
x
i
y
i
.
We will show that
y
defined this way is a bounded linear map. Linearity is easy
to see, and boundedness comes from the fact that
kyk
`
∗
p
= sup
x
ky(x)k
kxk
`
p
= sup
x
P
x
i
y
i
kxk
`
p
≤ sup
kxk
`
p
kyk
`
q
kxk
`
p
= kyk
`
p
,
by the H¨older’s inequality. So every (
y
i
)
∈ `
q
determines a bounded linear map.
In fact, we can show `
∗
p
is isomorphic to `
q
.