Riesz representation theorem (of linear functionals on function spaces)

Riesz representation theorem (of linear functionals on function spaces)

This entry should not be mistaken with the entry on the

Riesz representation theorem

of

bounded

(

http://planetmath.org/BoundedOperator

)

linear functionals

on an

Hilbert space

.

The Riesz provided here basically that linear functionals on certain

spaces of functions

can be seen as integration against

measures

. In other , for some spaces of functions all linear functionals have the form

f

⟼

∫

f

⁢

𝑑

μ

for some measure

μ

.

There are many versions of these Riesz , and which version is used depends upon the generality wishes to achieve, the difficulty of proof, the of space of functions involved, the of linear functionals involved, the of the ”” space involved, and also the of measures involved.

We present here some possible Riesz of general use.

Notation -

In the following we adopt the following conventions:

•

X

is a

locally compact Hausdorff space

.

•

C

c

⁢

(

X

)

denotes the space of real valued

continuous functions

on

X

with compact support.

•

C

0

⁢

(

X

)

denotes the space of real valued continuous functions on

X

that

vanish at infinity

.

•

all function spaces are endowed with the

sup-norm

∥

.

∥

∞

•

a linear functional

L

is said to be if

0

≤

L

⁢

(

f

)

whenever

0

≤

f

.

Theorem

1 (Riesz-Markov) -

Let

L

be a

positive linear functional

on

C

c

⁢

(

X

)

. There exists a unique

Radon measure

μ

on

X

, whose underlying

σ

-algebra (

http://planetmath.org/SigmaAlgebra

) is the

σ

-algebra

generated by

all

compact sets

, such that

L

⁢

(

f

)

=

∫

X

f

⁢

𝑑

μ

Moreover,

μ

is finite if and only if

L

is bounded.

Notice that when

X

is

σ

-

compact

(

http://planetmath.org/SigmaCompact

) the underlying

σ

-algebra for these measures is precisely the Borel

σ

-algebra (

http://planetmath.org/BorelSigmaAlgebra

) of

X

.

Theorem 2 -

Let

L

be a positive linear functional on

C

0

⁢

(

X

)

. There exists a unique finite Radon measure

μ

on

X

such that

L

⁢

(

f

)

=

∫

X

f

⁢

𝑑

μ

Theorem 3 (Dual of

C

0

⁢

(

X

)

) -

Let

L

be a linear functional on

C

0

⁢

(

X

)

. There exists a unique finite signed (

http://planetmath.org/SignedMeasure

)

Borel measure

on

X

such that

L

⁢

(

f

)

=

∫

X

f

⁢

𝑑

μ

0.0.1

Complex version:

Here

C

0

⁢

(

X

)

denotes the space of complex valued continuous functions on

X

that vanish at infinity.

Theorem 4 -

Let

L

be a linear functional on

C

0

⁢

(

X

)

. There exists a unique finite complex Borel measure

μ

on

X

such that

L

⁢

(

f

)

=

∫

X

f

⁢

𝑑

μ

Title

Riesz representation theorem (of linear functionals on function spaces)

Canonical name

RieszRepresentationTheoremofLinearFunctionalsOnFunctionSpaces

Date of creation

2013-03-22 17:28:18

Last modified on

2013-03-22 17:28:18

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

13

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 28C05

Classification

msc 46A99

Defines

Riesz-Markov theorem

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