nLab Riesz representation theorem

Summary

There are various related theorems in functional analysis and measure theory stating, under appropriate conditions, that the topological linear duals of various familiar Banach spaces (or something similar) are other familiar Banach spaces. Most of these are due in part to Frigyes Riesz, and many of them are named after him. Here we will consider them all together.

Throughout, we use notation for integrals in which unnecessary ‘$\mathrm{d}$’s are dropped; see the discussion on notation at measure space.

$C_c^{{*}{+}} = \overline{RM}^+$

Let $X$ be a locally compact Hausdorff space. Let $C_c(X)$ be the space of continuous functions on $X$ (valued in the complex numbers) with compact support; make $C_c(X)$ into a locally convex space with the topology of uniform convergence on compact subsets?; the dual vector space $C_c(X)*$ of this is (of course) the space of continuous linear functionals on $C_c(X)$; and the positive cone $C_c(X)^{{*}{+}}$ of this is the space of positive linear functionals on $C_c(X)$. Let $RM(X)$ be the space of finite Radon measures on $X$; make $RM(X)$ into a Banach space with the total variation? norm; the extended positive cone $\overline{RM(X)}^+$ of this is the space of positive Radon measures on $X$. Integration gives a map from $\overline{RM(X)}^+$ to $C_c(X)^{{*}{+}}$:

$$\mu \mapsto (f \mapsto \int_X f \mu) .$$

Theorem (Riesz)

This map is a homeomorphism:

$$C_c(X)^{{*}{+}} \cong \overline{RM(X)}^+ .$$

$C_0^* = RM$

Let $X$ be a locally compact Hausdorff space. Let $C_0(X)$ be the space of continuous functions on $X$ (valued in the complex numbers) on the one-point compactification of $X$ (so vanishing ‘at infinity’); make $C_0(X)$ into a Banach space with the supremum norm. Let $RM(X)$ be the space of finite Radon measures on $X$; make $RM(X)$ into a Banach space with the total variation? norm. Integration gives a map from $RM(X)$ to the dual vector space $C_0(X)^*$ of $C_0(X)$:

$$\mu \mapsto (f \mapsto \int_X f \mu) .$$

Theorem (Riesz–Markov)

This map is an isometric isomorphism:

$$C_0(X)^* \cong RM(X) .$$

$L_1^* = L_0$

$L_p^* = L_{1 - p}$

$H^* = \bar{H}$

$L_0^* = BA$

Related concepts

• GNS construction

References

A proof of Theorem in constructive mathematics (in the case where $X$ is a compactum) is given in

• Thierry Coquand, Bas Spitters, Integrals and Valuations (arXiv:0808.1522)