bounded linear functionals on L(μ)

For any measure spaceMathworldPlanetmath (X,𝔐,μ) and gL1(μ), the following linear map can be defined


It is easily shown that Φg is boundedPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath (, so is a member of the dual spaceMathworldPlanetmathPlanetmath of Ł(μ). However, unless the measure space consists of a finite setMathworldPlanetmath of atoms, not every element of the dual of L(μ) can be written like this. Instead, it is necessary to restrict to linear maps satisfying a bounded convergence property.


Let (X,M,μ) be a σ-finite ( measure space and V be the space of bounded linear maps Φ:L(μ)R satisfying bounded convergence. That is, if |fn|1 are in L(μ) and fn(x)0 for almost every xX, then Φ(fn)0.

Then gΦg gives an isometric isomorphism from L1(μ) to V.


First, the operator norm Φg is equal to the L1-norm of g (see Lp-norm is dual to Lq (, so the map gΦg gives an isometric embedding from L1 into the dual of L. Furthermore, dominated convergence implies that Φg satisfies bounded convergence so ΦgV. We just need to show that gΦg maps onto V.

So, suppose that ΦV. It needs to be shows that Φ=Φg for some gL1. Defining an additive set function ( ν:𝔐 by


for every set A𝔐, the bounded convergence property for Φ implies that ν is countably additive and is therefore a finite signed measure. So, the Radon-Nikodym theoremMathworldPlanetmath gives a gL1 such that ν(A)=Ag𝑑μ for every A𝔐. Then, the equality


is satisfied for f=1A with any A𝔐 and the functional monotone class theorem extends this to any bounded and measurable f:X, giving Φg=Φ. ∎

Title bounded linear functionalsMathworldPlanetmath on L(μ)
Canonical name BoundedLinearFunctionalsOnLinftymu
Date of creation 2013-03-22 18:38:08
Last modified on 2013-03-22 18:38:08
Owner gel (22282)
Last modified by gel (22282)
Numerical id 5
Author gel (22282)
Entry type Theorem
Classification msc 28A25
Related topic BoundedLinearFunctionalsOnLpmu
Related topic RadonNikodymTheorem
Related topic LpNormIsDualToLq