bounded linear functionals on L^p⁢(μ)

bounded linear functionals on

L

p

⁢

(

μ

)

If

μ

is a

positive measure

on a set

X

,

1

≤

p

≤

∞

, and

g

∈

L

q

⁢

(

μ

)

, where

q

is the Hölder conjugate of

p

, then Hölder’s

inequality

implies that the map

f

↦

∫

X

f

⁢

g

⁢

𝑑

μ

is a

bounded

linear functional

on

L

p

⁢

(

μ

)

. It is therefore natural to ask whether or not all such

functionals

on

L

p

⁢

(

μ

)

are of this form for some

g

∈

L

q

⁢

(

μ

)

. Under fairly mild hypotheses, and excepting the case

p

=

∞

, the

Radon-Nikodym Theorem

answers this question affirmatively.

Theorem

.

Let

(

X

,

M

,

μ

)

be a

σ

-

finite measure space

,

1

≤

p

<

∞

, and

q

the Hölder conjugate of

p

. If

Φ

is a bounded linear functional on

L

p

⁢

(

μ

)

, then there exists a unique

g

∈

L

q

⁢

(

μ

)

such that

Φ

⁢

(

f

)

=

∫

X

f

⁢

g

⁢

𝑑

μ

(1)

for all

f

∈

L

p

⁢

(

μ

)

. Furthermore,

∥

Φ

∥

=

∥

g

∥

q

. Thus, under the stated hypotheses,

L

q

⁢

(

μ

)

is

isometrically isomorphic

to the

dual space

of

L

p

⁢

(

μ

)

.

If

1

<

p

<

∞

, then the assertion of the

theorem

remains valid without the

assumption

that

μ

is

σ

-finite; however, even with this

hypothesis

, the result can fail in the case that

p

=

∞

. In particular, the bounded linear functionals on

L

∞

⁢

(

m

)

, where

m

is

Lebesgue measure

on

[

0

,

1

]

, are not all obtained in the above manner via members of

L

1

⁢

(

m

)

. An explicit example illustrating this is constructed as follows: the assignment

f

↦

f

⁢

(

0

)

defines a bounded linear functional on

C

⁢

(

[

0

,

1

]

)

, which, by the

Hahn-Banach Theorem

, may be extended to a bounded linear functional

Φ

on

L

∞

⁢

(

m

)

. Assume for the sake of

contradiction

that there exists

g

∈

L

1

⁢

(

m

)

such that

Φ

⁢

(

f

)

=

∫

[

0

,

1

]

f

⁢

g

⁢

𝑑

m

for every

f

∈

L

∞

⁢

(

m

)

, and for

n

∈

ℤ

+

, define

f

n

:

[

0

,

1

]

→

ℂ

by

f

n

⁢

(

x

)

=

max

⁡

{

1

-

n

⁢

x

,

0

}

. As each

f

n

is

continuous

, we have

Φ

⁢

(

f

n

)

=

φ

⁢

(

f

n

)

=

1

for all

n

; however, because

f

n

→

0

almost everywhere and

|

f

n

|

≤

1

, the

Dominated Convergence Theorem

, together with our hypothesis on

g

, gives

1

=

lim

n

→

∞

⁡

Φ

⁢

(

f

n

)

=

lim

n

→

∞

⁡

∫

[

0

,

1

]

f

n

⁢

g

⁢

𝑑

m

=

0

⁢

,

a contradiction. It follows that no such

g

can exist.

Title

bounded linear functionals on

L

p

⁢

(

μ

)

Canonical name

BoundedLinearFunctionalsOnLpmu

Date of creation

2013-03-22 18:32:57

Last modified on

2013-03-22 18:32:57

Owner

azdbacks4234 (14155)

Last modified by

azdbacks4234 (14155)

Numerical id

15

Author

azdbacks4234 (14155)

Entry type

Theorem

Classification

msc 28B15

Related topic

LpSpace

Related topic

HolderInequality

Related topic

ContinuousLinearMapping

Related topic

BanachSpace

Related topic

DualSpace

Related topic

ConjugateIndex

Related topic

RadonNikodymTheorem

Related topic

BoundedLinearFunctionalsOnLinftymu

Related topic

LpNormIsDualToLq

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