nLab
Riesz representation theorem

The Riesz representation theorems

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 ‘d\mathrm{d}’s are dropped; see the discussion on notation at measure space.

C c *+=RM¯ +C_c^{{*}{+}} = \overline{RM}^+

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

μ(f Xfμ). \mu \mapsto (f \mapsto \int_X f \mu) .
Theorem (Riesz)

This map is a homeomorphism:

C c(X) *+RM(X)¯ +. C_c(X)^{{*}{+}} \cong \overline{RM(X)}^+ .

C 0 *=RMC_0^* = RM

Let XX be a locally compact Hausdorff space. Let C 0(X)C_0(X) be the space of continuous functions on XX (valued in the complex numbers) on the one-point compactification of XX (so vanishing ‘at infinity’); make C 0(X)C_0(X) into a Banach space with the supremum norm?. Let RM(X)RM(X) be the space of finite Radon measures on XX; make RM(X)RM(X) into a Banach space with the total variation? norm. Integration gives a map from RM(X)RM(X) to the dual vector space C 0(X) *C_0(X)^* of C 0(X)C_0(X):

μ(f Xfμ). \mu \mapsto (f \mapsto \int_X f \mu) .
Theorem (Riesz–Markov)

This map is an isometric isomorphism?:

C 0(X) *RM(X). C_0(X)^* \cong RM(X) .

L 1 *=L 0L_1^* = L_0

L p *=L 1pL_p^* = L_{1 - p}

H *=H¯H^* = \bar{H}

L 0 *=BAL_0^* = BA

References

A proof of Theorem 1 in constructive mathematics (in the case where XX is a compactum) is given in

Revised on August 22, 2012 21:43:03 by Urs Schreiber (89.204.139.7)