Riesz-Kakutani Representation Theorem

Theorem

Let $X$ be a Hausdorff compact space.

Let $\map \BB X$ be the Borel $\sigma$-algebra on $X$.

Let $\map C {X, \R}$ be the space of real-valued continuous functions.

Let $\norm {\, \cdot \,}_\infty$ be the supremum norm on $X$.

Let $\struct {C', \norm {\, \cdot \,}_{C'} }$ be the normed dual space of $\struct {\map C {X, \R}, \norm {\, \cdot \,}_\infty}$.

Then $\ell \in C'$ if and only if there is a unique signed measure $\mu$ on $\struct {X, \map \BB X}$ such that:

the variation $\size \mu$ of $\mu$ is finite

$\ds \forall f \in \map C {X, \R} : \map \ell f = \int_X f \rd \mu$

In addition:

$\norm \ell_{C'} = \map {\size \mu} X$

Proof

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Source of Name

This entry was named for Frigyes Riesz and Shizuo Kakutani.

Sources

• 2002: Peter D. Lax: Functional Analysis: $8.3$: Reflexive Spaces