M. Riesz extension theorem

The M. Riesz extension theorem is a theorem in mathematics, proved by Marcel Riesz during his study of the problem of moments.

Formulation

Let

E

{\displaystyle E}

be a real vector space,

F

⊂

E

{\displaystyle F\subset E}

be a vector subspace, and

K

⊂

E

{\displaystyle K\subset E}

be a convex cone.

A linear functional

ϕ

:

F

→

R

{\displaystyle \phi :F\to \mathbb {R} }

is called

K

{\displaystyle K}

-positive, if it takes only non-negative values on the cone

K

{\displaystyle K}

:

ϕ

(

x

)

≥

0

for

x

∈

F

∩

K

.

{\displaystyle \phi (x)\geq 0\quad {\text{for}}\quad x\in F\cap K.}

A linear functional

ψ

:

E

→

R

{\displaystyle \psi :E\to \mathbb {R} }

is called a

K

{\displaystyle K}

-positive extension of

ϕ

{\displaystyle \phi }

, if it is identical to

ϕ

{\displaystyle \phi }

in the domain of

ϕ

{\displaystyle \phi }

, and also returns a value of at least 0 for all points in the cone

K

{\displaystyle K}

:

ψ

|

F

=

ϕ

and

ψ

(

x

)

≥

0

for

x

∈

K

.

{\displaystyle \psi |_{F}=\phi \quad {\text{and}}\quad \psi (x)\geq 0\quad {\text{for}}\quad x\in K.}

In general, a

K

{\displaystyle K}

-positive linear functional on

F

{\displaystyle F}

cannot be extended to a

K

{\displaystyle K}

-positive linear functional on

E

{\displaystyle E}

. Already in two dimensions one obtains a counterexample. Let

E

=

R

2

,

K

=

{

(

x

,

y

)

:

y

>

0

}

∪

{

(

x

,

0

)

:

x

>

0

}

,

{\displaystyle E=\mathbb {R} ^{2},\ K=\{(x,y):y>0\}\cup \{(x,0):x>0\},}

and

F

{\displaystyle F}

be the

x

{\displaystyle x}

-axis. The positive functional

ϕ

(

x

,

0

)

=

x

{\displaystyle \phi (x,0)=x}

can not be extended to a positive functional on

E

{\displaystyle E}

.

However, the extension exists under the additional assumption that

E

⊂

K

+

F

,

{\displaystyle E\subset K+F,}

namely for every

y

∈

E

,

{\displaystyle y\in E,}

there exists an

x

∈

F

{\displaystyle x\in F}

such that

y

−

x

∈

K

.

{\displaystyle y-x\in K.}

Proof

The proof is similar to the proof of the Hahn–Banach theorem (see also below).